Thesis Manuscript - Materials Physics Center

Transcription

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Department of Physics
Grupo de Fı́sica Teórica de la Materia Condensada
Universidad de los Andes
Plasmon resonances and tunneling in
sub-nanometric systems
Thesis developed by
MARIO ZAPATA HERRERA
for the degree of
Doctor en Ciencias Fı́sica
Supervised by
Dr. Ret. Nat. Angela Stella Camacho Beltrán
Bogotá-Colombia, December 2015.
“Muchos años después, frente al pelotón de fusilamiento, el coronel Aureliano Buendı́a
habı́a de recordar aquella tarde remota en que su padre lo llevó a conocer el hielo.
Macondo era entonces una aldea de veinte casas de barro y cañabrava construidas a
la orilla de un rı́o de aguas diáfanas que se precipitaban por un lecho de piedras pulidas,
blancas y enormes como huevos prehistóricos. El mundo era tan reciente, que muchas
cosas carecı́an de nombre, y para mencionarlas habı́a que señalarlas con el dedo. Todos los años, por el mes de marzo, una familia de gitanos desarrapados plantaba su
carpa cerca de la aldea, y con un grande alboroto de pitos y timbales daban a conocer
los nuevos inventos. Primero llevaron el imán. Un gitano corpulento, de barba montaraz
y manos de gorrión, que se presentó con el nombre de Melquı́ades, hizo una truculenta demostración pública de lo que él mismo llamaba la octava maravilla de los sabios
alquimistas de Macedonia...
Al ser destapado por el gigante, el cofre dejó escapar un aliento glacial. Dentro sólo
habı́a un enorme bloque transparente, con infinitas agujas internas en las cuales, se
despedazaba en estrellas de colores la claridad del crepúsculo. Desconcertado, sabiendo
que los niños esperaban una explicación inmediata, José Arcadio Buendı́a se atrevió a
murmurar:
—Es el diamante más grande del mundo.
—No —corrigió el gitano—. Es hielo.
José Arcadio Buendı́a, sin entender, extendió la mano hacia el témpano, pero el gigante
se la apartó. “Cinco reales más para tocarlo”, dijo. José Arcadio Buendı́a los pagó, y
entonces puso la mano sobre el hielo, y la mantuvo puesta por varios minutos, mientras el corazón se le hinchaba de temor y de júbilo al contacto del misterio. Sin saber
qué decir, pagó otros diez reales para que sus hijos vivieran la prodigiosa experiencia.
El pequeño José Arcadio se negó a tocarlo. Aureliano, en cambio, dio un paso hacia
adelante, puso la mano y la retiró en el acto. “Está hirviendo”, exclamó asustado. Pero
su padre no le prestó atención. Embriagado por la evidencia del prodigio, en aquel momento se olvidó de la frustración de sus empresas delirantes y del cuerpo de Melquı́ades
abandonado al apetito de los calamares. Pagó otros cinco reales, y con la mano puesta
en el témpano, como expresando un testimonio sobre el texto sagrado, exclamó:
—Este es el gran invento de nuestro tiempo.”
Gabriel Garcı́a Márquez. Cien Años de Soledad
Dedicada a mis padres Magdala y Mario
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Acknowledgements
The quest for knowledge from the mathematical description of Nature, for me, is one of
the most beautiful adventures in the complexity of mankind and in its most fundamental
version, known as physics, one of the disciplines of greatest acceptance and impact on
western society, at least since the time of Galileo. This makes acknowledging in general
to the whole human race for having accomplished to the end of this thesis, would be
just a humble pretentiousness, from the tiny glance of a human being, fellow of the trip
in this spaceship called Earth that is glimpsed from the vast universe as a “pale blue
dot”, in Carl Sagan’s words. However, “the others”, those to whom we called relatives,
family or friends have gone along with nearby each story of life, in this infinite universe
manifeste within us, in this journey called existence. To them, who are just oneself in
another skin, they are to whom simply I give my sincere thanks for being the architects
of each one of the letters, equations and results in this humble and honest interpretation
of this broad universe.
First, I would to express my gratitude to God and my parents, for being the squires of
my existence with their guidance, advices and for showing me that the freedom of being
myself in every challenge, not only in academy, is the key to keep the horizon so far,
though with the capacity of seeing it always close. I render thanks to God for them,
to Magdala and Mario for being my parents, who accepted my choice of having them
as family, alongside my siblings: Marta, James, Emma, Yuri, my lovely sister Marcela
and my nephews and nieces. For their love and unconditional devotion, Thank you very
much!.
On the other hand, I want to thank infinitely to my adviser and friend Angela Camacho. I remember thousands of times that I was not conformed at all and indifferent to
everything, and your response was as mother’s every time, patiently and a lot of love
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in an ambient that for many reasons sometimes it turns cold. Angela: long life to you
and thanks for giving so much to your country, to physics, to the University, to women
scientist of the world and in special to your students. You are a fundamental part of this
process.
To my dearest friends, who with their love, and despite infinite mistakes they have
borne me, they deserve this starry sky that always welcome us in endless discussions,
long “parrandas” and countless stories of life, poetry and science, searching a better
road, a fairer world. Thanks for making of this road something sincere, a road worth
to live. To Cesar Herreño, Nicolás Ávilan, Juan Carlos Arias, Julian Jimenez, Camilo
Moscoso, Javier Cano, Manuel Torres, simply thanks a lot and hope to see you the rest
of my life, because you have been great when I have been small in skimpy and fateful
times, in Fito’s words, “En tiempos donde siempre estamos solos”. Thanks for being
always there and giving me a lot of my life, for being great masters and being the best
friends anyone can have.
To the “Pecho y Biceps”: Larry Castrillon, Jorge Sánchez (muscle), JuanPa Mallarino,
Alejandro Ferrero and Alejandro Mahecha, thanks for joining the last year full of stories
and for always being ready to lift me up every time I was down. I will never be able
to repay all you have done for me. To my first fellows in my first year of Ph.D: Felipe
Mendez, Felipe Gonzalez, Nathaly Marı́n, Oscar Acevedo, Juan Carlos Pasto, Giovany
Ruiz y Jefferson Florez, thanks for making of Uniandes my second home. To my plenitys:
César Rock and Amanda, Alvarito, Dayana, Luisa Caviedes, Chema and in special to
John Mora and Carolina Jimenez. Thanks for your sincerity, love, rational and irrational
discussions about life, rock, women and in special for giving me a great sarcastic and
definitive laugh where it was required. Thanks always for the music and giving me a
home.
A special place is for Edwin Herrera: I only want to say that God give me the opportunity to know not only a friend, but a brother. Thank you for be my confidant, my best
friend and for all that great light and even dark moments that we joined in this life.“
Gracias Parcero”.
Acknowledgement to those friends who appear out of nowhere, when everything seemed
to be lost. Thanks friends BAOBAB and people from “Accesa Centro de Liderazgo”,
in special to Andrea Cortés, Andrea Camacho, Yamile Casallas, Helen Gorder, to my
coach and great “santafereño” Holman Borda. Thanks for the gift of love and for learn
me to “trust” and for playing the game “Ganar Ganar”. A heartfelt “4” for everyone.
To my friends from Quimbaya: my soul friends from the Banda Municipal, specially my
brother Kurt and Báquiro (baby!). Dorita and Maria Antonia: you have a part of my
heart. Thanks to Aleja Salazar, Sandra Velez and Orlando Rojas. Also my foster parents
in Armenia Sandra Vera and Julian Ramirez: “un dios les pague” for this last months
in Quindı́o writing my thesis. Also special thanks to Alejandra and Santi Granada for
letting me use their great “café de altura” as an office.
Agradecimientos especiales a Jhuliana Rojas por ser parte de mi vida en momentos de
mucha felicidad y amor. Gracias a tus hijas Sarita y Paulita. Gracias por poyarme en
todo momento y por creer en mi.
I want to thank to my family, in special my grandparents Octavio and Franquelina, uncle
Arturo and Blanquita, uncle Gilberto (Mono) and my cousin Lina Riveros, uncle Diego
and very deeply to my aunt Mary and her husband Germán and my cousin Dianita
(primeins) for their support and make me at home when I did not have one. Thanks
for making of Tofi the first dog, at least in the family, to be part in a doctoral thesis.
To Claudina Matinez and Eduardo Muñoz: You have a special part of my heart, as well
as Nico, Natalia and my dear cousin Mona (Monica Melissa). No words can express my
deep feelings and love I have for you, and how beautiful it is to write these lines knowing
you are part of my life as two squires, despite of my imperfections as human being.
Thanks to my uncle Benji and my aunt Patty in Madrid: You give me the best moments
and turism in Spain.
Thanks to professor Camilo Espejo and Hanz Ramirez. You were very kind to share with
me codes, methods, bibliography and very valuable help in several academic and spiritual
discussions. Likewise to Universidad de los Andes, the Facultad de Ciencias for financial
support through all my “Proyectos Semillas”. To the directors of the Department of
Physics during my PhD program, Carlos Avila and Gabriel Tellez and all the administrative staff: Johana, William Paiba, Julieta Rodriguez and Luis Anibal. Special thanks
to Rubén Esteban for introducing me in the QCM calculations and Codruta Marinica for
sharing me codes, information and very productive conversations in Donosti and Paris.
To the people of Centro de Fı́sica de Materiales and the Donostia International Physics
Center in the Basque Country, thanks fot financial and academic support. I made there
the major part of this work. Thanks for facilitate my stay during three visits in less than
two years.
To my kindly friends in “la bella Easo”: my dear “ser de luz”Manuel Blanco, thanks for
be the link with my other friends there and invite me to be a part of the“’cuadrilla”.
Thanks to Christian Fernández (Cheito), Camilo Cortés, Lina Gutierrez, Johana Brito,
David Amico, Wissem (el Túnez), Ammen Poyli, Holanda Gomez and Sarita Quin for
making my stay one of the most wonderful experiences in my life. Very good memories
to Salomé Valencia and Karen Alonso, thanks for your lovely friendship, help and for
visit me. To Nena (Miryam Helena López), Nana, Jorge Hugo Ossa, Nicolás Esquivel,
Jorge Molina and Andrés Pitarra in Paris. Merci beaucoup!
Before concluding, I would like to express my gratitude to all of colombian people who
pay honestly their taxes. You make possible to people like me to accomplish science
education, in a country where unfortunately, science is not funded like army and fight
against war. Thanks to COLCIENCIAS for supporting this doctoral program. I can assure you that the money was invested in the best hands.
Last but not least important, I would like to dedicate these last lines to two of the
most important people I have ever known in my history as a physicist. Lehenik eta
behin, Javier Aizpuruari, zeinak bere adeitasun, konfiantza eta ardurarekin posible egin
baitzuen nire egonaldia Nanofotonikako taldean, Donostiako Materialen Fisikarako Zentroan. Mila esker ezagutzen ez zenuen norbaitengan konfiantza izatearren eta beti ongi etorria den eskua luzatzearren; fisikari buruzko eztabaidengatik eta nire kezka eta
galderentzat beti tarte bat aurkitzearren; munduan nanoplasmonika eta nanozientzia
gara daitezen laguntzearren, nigan konfiantza izatearren, betidanik parte izan nahi nuen
mundu bat ezagutaraztearren, eta beste behin ere karrera miresgarri honetan sinesteko
aukera oparitzearren. Mila esker, bihotzez.
Finally, I want not only say thanks, but praising the great human spirit of Andrei Borisov,
to whom I met in Donosti during my first visit to CFM and ISMO and from whom I
got the best image of an universal man, from a human being who transcends language,
cultural and social frontiers in honor of knowledge and reason. From you, dear Andrei,
I hope I have not learnt everything yet, I hope to keep your support, long discussions
on science, Europe and about the world. Thanks for being my (non-official) adviser and
for making me fell in love with physics and especially because I will never forget your
words of encouragement in moments of great despair: “I can tell you is that is what you
have to keep on going, and one day the life will be full of colors once again. And with
little chance you might decide that somebody was right above forcing you to go-through
what you are going now. Well, in any case, we have to believe that anything happens
that our life will drive onto a better road. You have so many friends. This should help”.
Now, Andrei, the world is full of colors. You were completly right one more time. Thank
you.
Mario Zapata Herrera,
Quimbaya, Colombia.
Bogotá, Colombia.
Donostia-San Sebastián, Spain. December 2015
Abstract
One of the most promising and most active nowadays research field in nanoscience,
is the field of Nano-Plasmonics, which describes the light interaction with nanostructured metallic systems. The optical response of these nanostructures is determined by
a collective resonance of conduction electrons which depends on geometrical and compositional factors as well as on the frequency and polarisation properties of the incident
light. For large systems of tens of nanometers in size and above, this response is well
understood, and can be confidently described by classical Maxwell equations. At the
same time, for plasmonic nanoparticles of some nm size and below, or for plasmonic
nanostructures characterised by narrow subnanometric gaps between nanoparticles, the
quantum effects due to the non-local screening and tunneling emerge. These quantum
effects are not captured by standard classical descriptions which is then a real challenge
for the theory. Indeed, (i) Nowadays technology allows controllable engineering of such
structures which asks for theoretical predictions; (ii) Quantum effects can strongly affect
the plasmonic resonances of a hybrid nanostructure as has been demonstrated for e.g.
small metallic clusters, plasmonic dimers and core-shell nanoparticles. Thus, effective
electron tunneling across these gaps alters both the far-field optical response and the
near-field confinement and enhancement. While generally, quantum descriptions using
full quantum mechanical calculations like Time Dependent Density Functional Theory
(TDDFT) are needed to account for quantum effects in subnanometric gaps, recently
developed so-called Quantum Corrected Model (QCM) [1, 2] allows to incorporate electron tunneling within the framework of classical Maxwell equations. In this approach,
the gap is filled with an effective medium characterized by a tunneling conductance. The
advantage of this methodology is that it allows to tackle large (realistic) plasmonic systems with millions of atoms which is at present out of reach for the quantum descriptions.
xi
In this thesis we have theoretically studied the optical response of several canonical systems with narrow gaps: nanoparticle pairs (dimers), and core-shell systems (known in
literature as nanomatryoshkas), in the subnanometric regime (systems below 10 nm). As
a first approach, we have used classical electromagnetics treatment allowing a link with
previous studies and characterisation of the plasmonic modes and near and far fields. We
have explored the physics of the plasmon coupling and near field enhancements in plasmonic cavities upon reduction of the gap size, where classical theory predicts extremely
strong effects because of the attractive Coulomb interaction between plasmon induced
charges across the junction [3]. For the vanishing width of the junction this interaction
diverges leading to the discontinuities in classical theory [4]. This is precisely the regime
where quantum effects like tunneling are of paramount importance as they neutralise
the screening charges and allow continuous transition from the capacitive to inductive
coupling at zero gap size (continuous metal in the junction).
In a joint experimental and theoretical study (in a work performed in collaboration with
an experimental group1 ), we have investigated the plasmon coupling in gold nanomatryoshkas with different core-shell separations. In agreement with experimental data the
effects of the hybridization between the core and the shell plasmons become significant
for the gap widths below 15 nm. When the gap width decreases to below 1 nm, the nearand far-fields can no longer be described by classical approaches but require the inclusion of quantum mechanical effects such as electron transport through the self-assembled
molecular monolayer present in the junction. Surface-Enhanced Raman Scattering measurements indicate strong effects because of the charge transfer across the molecular
junction. In this respect, we found out that quantum modeling provides an estimate for
the AC conductance of molecules in the junction. The insights acquired from this part
of the work, pave the way for the development of novel quantum plasmonic devices and
substrates for Surface-Enhanced Raman Scattering (SERS) [5].
In the combined effort with research groups from Spain 2 and France 3 , we have proposed
a mechanism for fast and active control of the optical response of metallic nanostructures, based on exploiting the quantum regime of subnanometric plasmonic gaps. We
1
Jian Ye Group. School of Biomedical Engineering and Med-X Research Institute, Shanghai Jiao
Tong University.
2
Javier Aizpurua, Nanophotonics Group. Materials Physics Center and Donostia International
Physics Center. Donostia-San Sebastian-Spain.
3
Andrei. G. Borisov. Institut des Sciences Moléculaires d’Orsay, Université Paris Sud, Paris-France.
demonstrated that by applying a DC bias across a narrow gap in core-shell nanostructures (cylindrical nanomatryoshkas) a substantial change of the tunneling conductivity
across the junction can be induced at optical frequencies. In turns this modifies in a
reversible manner the plasmonic resonances of the core shell systems. We demonstrate
the feasibility of the concept of active control using time dependent density functional
theory calculations. Thus, along with two-dimensional structures, metal nanoparticle
plasmonics can benefit from the reversibility, fast response times, and versatility of an
active control strategy based on applied bias. The proposed electrical manipulation of
light using quantum plasmonics, establishes a new platform for many practical applications in optoelectronics [6].
Finally, using time dependent density functional theory (TDDFT) we performed the
full quantum calculation of the optical response, stability, and electron loss for charged
spherical clusters. We have obtained that, because the system stays neutral in the bulk,
and the extra-charge is localized at the surface of the cluster, the dipolar plasmon mode
displays only small frequency shift linked with change of the electron spill-out from the
nanoparticle boundaries [7]. We have also shown that even small (∼ 10 % relative to the
total number of electrons in the system) negative charge raises Fermi level of the cluster
above the vacuum level and renders the system unstable. The extra charge of the cluster
decays with characteristic time constants that can be fully understood using an analytical
study based on the WKB method, similar to the studies of the alpha-decay. Our results
on the charged clusters allow to critically reinterpreting many of the experimental data obtained with electrochemistry, where the shift of the plasmon resonances has been
attributed to the charging effects. Based on our results, we tentatively propose an alternative explanation based on the change of the dielectric environment in immediate
vicinity of the cluster.
Contents
Acknowledgements
V
Abstract
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Contents
XIII
1. Introduction
1.1. Plasmonics . . . . . . . . . . . . . . . . . . .
1.1.1. The dielectric function . . . . . . . . .
1.1.2. Plasmon resonances and nanoparticles
1.1.3. Near Field Enhacement Factor . . . .
1.1.4. Nanoparticle dimers and coupling . . .
1.2. Quantum Plasmonics . . . . . . . . . . . . . .
1.2.1. Active Quantum Plasmonics . . . . .
1.3. Summary . . . . . . . . . . . . . . . . . . . .
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2. Confinement potential effects on the Localized Surface Plasmon
Resonance of metallic small nanoparticles
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2. The dielectric function . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1. Free electron gas in a spherical infinite well
confinement: Exact solution . . . . . . . . . . . . . . . . . . .
2.2.2. Free electron gas in a spherical infinite well
confinement: Approximate solution . . . . . . . . . . . . . . .
2.2.3. Quantum finite confinement . . . . . . . . . . . . . . . . . . .
2.2.4. Confinement effects on the optical response . . . . . . . . . .
2.3. Absorption spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4. Enhancement of the near field . . . . . . . . . . . . . . . . . . . . . .
2.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3. Quantum effects in the optical response of core-shell
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . .
3.2. System and computational aspects . . . . . . . . . . .
3.2.1. Optical response in the TDDFT framework . .
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3.2.2. The quantum corrected model in core-shell nanomatryoshkas
3.3. Plasmon modes of a nanomatryoshka . . . . . . . . . . . . . . . . . .
3.3.1. Near field distribution . . . . . . . . . . . . . . . . . . . . . .
3.4. Quantum tunneling across the gap. . . . . . . . . . . . . . . . . . . .
3.4.1. The near field enhancement . . . . . . . . . . . . . . . . . . .
3.4.2. Plasmon modes of a spherical gold nanomatryoshka . . . . .
3.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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4. Tunneling effects in spherical gold nanomatryoshkas
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2. System and general considerations . . . . . . . . . . . . . . . . . . . .
4.3. Plasmon hybridization and quantum effects in NM of different sizes . .
4.3.1. Quantum tunneling in NM and subnanometric molecular layers
4.4. SERS measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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5. Active Quantum Plasmonics
5.1. Introduction . . . . . . . . . . . . . . . . . . . .
5.2. Model and qualitative aspects . . . . . . . . . .
5.3. TDDFT framework . . . . . . . . . . . . . . . .
5.3.1. Tunneling effects . . . . . . . . . . . . .
5.3.2. TDDFT approach for the systems under
5.4. Control with applied bias. . . . . . . . . . . . .
5.5. Quantum corrected model. QCM. . . . . . . .
5.6. Time dependence of the induced dipole . . . . .
5.7. Summary of main results and conclusions . . .
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6. Quantum Plasmonics and charged clusters dynamics
6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2. Model and computational aspects . . . . . . . . . . . . . . . . .
6.3. Ground state properties . . . . . . . . . . . . . . . . . . . . . .
6.4. Absorption cross-section . . . . . . . . . . . . . . . . . . . . . .
6.5. TDDFT study of the electron population decay in negatively
clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5.1. Analytical expressions . . . . . . . . . . . . . . . . . . .
6.5.2. TDDFT results . . . . . . . . . . . . . . . . . . . . . . .
6.6. Summary and Conclussions . . . . . . . . . . . . . . . . . . . .
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7. General Summary
7.1. Analytical and numerical methods . . . . . . . . . . . . . . . . . . . . .
7.2. Studied Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3. Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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A. Quantum model for a free electron gas in a nanoparticle
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A.1. Free electron gas in a cubic infinite box . . . . . . . . . . . . . . . . . . . 121
Abbreviations
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List of publications
151
Chapter 1
Introduction
One of the greatest advances in the description of the physical properties of materials in
the last century concerned an understanding of the properties of matter in the so-called
low dimensional systems. The interaction of light and matter has allowed probing the
deepest secrets of the physical nature of our universe, by analyzing the exchange of
energy and momentum between photons and the electrons of the materials. Furthermore, this interaction has helped to develop many technological applications that have
contributed to the spectacular technological development in the twentieth and twentyfirst centuries. In this context, the control and manipulation of light has been limited by
diffraction, which provides an intrinsic limit to the spatial location below the wavelength.
It is only recently that the advances in nanofabrication techniques made it possible to
design and built nanostructured materials allowing to manipulate light at the scales below the wavelength [8–10]. We present in this chapter the main generalities of a field of
plasmonics aimed at the fundamental and applied studies of the light interaction with
nanostructures supporting collective electronic excitations, plasmons.
1.1.
Plasmonics
Some nanomaterials have the ability to localize light below the diffraction limit due to
their electronic excitations. Specifically, metals exhibit a coherent collective excitation of
the conduction electrons, commonly called plasmon. When metals are structured in the
form of nanoparticles, these plasmons are localized in the dimensions of the nanoparticle, resulting in a distribution of the associated optical field, which beats the diffraction
1
Chapter 1
Introduction
Figure 1.1: Lycurgus cup. A dichroic glass cup with a mythological scene. Late Rome,
4th century AD. This extraordinary cup is the only complete example of a very special
type of glass, known as dichroic, which changes color when held up to the light. The
opaque green cup (left) turns to a glowing translucent red (right) when light is shone
through it. The glass contains tiny amounts of colloidal gold and silver, which give it
these unusual optical properties. The reason for this dichroism comes from plasmonic
effects. (Image and text: British Museum webpage http://www.britishmuseum.org/)
limit, and allows to locate and manipulate light in dimensions well below the wavelength1 . The nanolocalization of light attracts a fundamental interest, and moreover it
holds a great promise of numerous practical applications, which explains the outstanding
development of the field of plasmonics over last years.
Historically speaking, plasmon phenomena have been used by humanity since design
of the colors of the stained glass windows in medieval cathedrals, or even older, since
roman 4th century AD with beautiful example given by the Lycurgus cup (see Fig. 1.1).
However, the understanding of the underlying physics came latter with works of Gustav
Mie [11], who derived in the beginning of the twentieth century an analytical solutions
to the Maxwell equations in matter, describing the scattering and absorption of light
by spherical particles. The quantum description has been provided in works of Rufus
Ritchie [12] in the mid-twentieth century, who studied collective excitations of the conduction electrons in a free electron metal using a hydrodynamical model (HDM). This
kind of excitations have been an important part of the study of the optical response of
1
In this work, we only refer to localized surface plasmons in metallic nanoparticles. There is another
kind of plasmon resonances refered to surface propagating plasmon excitations, which are not subject of
this thesis.
2
1.1
Plasmonics
matter under the incidence of electromagnetic waves [13]. Nowadays, research in plasmonics is aimed at the technological applications, mainly on exploiting the enhancement
of the optical fields [14–16] in the metal nanoparticles (or near field enhancement) vicinity. Thus, significant developments are achieved in the field of sensor technology [17, 18],
nanoantennas for single molecule detection [19], medical research [20], optical information transfer [21], single photon generation [22], control and generation of nonlinear
effects [23], photochemistry [24], heat generation and injection of hot electrons [25],
that facilitate among others, chemical reactions. The technological progress makes it
possible to produce the characteristic geometries of plasmonic systems such that the
quantum effects start to be important. Because of the tunneling and non-local screening
the classical electromagnetic theory based on the solution of Maxwell equations with
local dielectric description of the media cannot be applied. Quantum treatment and
modeling are needed in this case, which stimulated efforts in full quantum calculations
(like TDDFT), or in simplified effective models (like QCM) [1, 2, 26, 27] in order to
calculate the optical response of metallic nanoparticles [28].
1.1.1.
The dielectric function
Within the classical approach, the plasmon phenomena are addressed with the solution of
electrodynamical Maxwell’s equations. The materials are characterised by the dielectric
function ε. The sharp material boundaries are typically assumed, and the spatial dispersion of the material response is neglected so that only the frequency-dependence of
the dielectric function is accounted for ε = ε(ω).
~ r, t) and the electric field E(~r, t) of an isotropic, linear and
The displacement vector D(~
homogeneous material are related in the reciprocal space (~k, ω) by a constitutive equation,
including the dielectric function ε(ω), which is given by
~ ~k, ω) = ε(ω)E(
~ ~k, ω),
D(
(1.1)
where the local dielectric function of the metal ε(ω) can be obtained either from empirical
data or theoretically approximated with e.g. the well-known Drude Model (DM). In this
model, one considers the conduction electrons as a free electron gas with a density n of
charges per unit volume moving on a background of positive ion cores. The electrons are
non-interacting within this model and all the information about realistic band structure
3
Chapter 1
Introduction
is contained in the electron effective mass me [13]. Starting with a classical equation
of motion for an electron in a plasma under the influence of an external electric field
~
E(t)
= E~0 e−iωt , and assuming a harmonic response, the dielectric function of a metal
can be obtained in the following form
ωp2
.
ω 2 + iγω
ε(ω) = ε∞ −
(1.2)
In Eq.(1.2) ∞ represents the dielectric screening due to the interband transitions (∞ =
1 for vacuum).
s
ωp =
ne2
ε0 m
(1.3)
is the bulk plasma frequency defined in terms of the electron density n, the electron
charge e, the electron effective mass m and the free space electric permittivity ε0 . Finally, γ is the excitation damping rate, which is a phenomenological quantity that accounts many of the posible electron interactions and allows us to do classic or quantum
approaches. Eq.(1.2) gives a good description of dielectrical properties of metals in bulk
and in large nanoparticles. In small metal nanoparticles with a very high surface to
volume ratio, surface scattering of conduction electrons leads to the increased plasmon
damping rate. The surface contribution can be accounted for in descriptions of the plasmon response of small nanoparticles modifying the damping term in Eq.(1.2) γ → γ(R)
[29], where R is the mean nanoparticle radius, and
γ(R) = γBulk +
AvF
.
R
(1.4)
Here vF is the metal Fermi velocity and A is a geometrical parameter.
Using Eq.(1.2) (taking ε∞ = 1), we can separate the dielectric functon into the real
ε1 (ω) and imaginary ε2 (ω) parts.
ωp2 τ 2
1 + ω2τ 2
(1.5)
ωp2 τ
.
ω(1 + ω 2 τ 2 )
(1.6)
ε1 (ω) = 1 −
and
ε2 (ω) =
4
1.1
Plasmonics
The parameter τ = 1/γ corresponds to the relaxation time of the free electron gas τ ,
given by the inverse of the collision frequency γ from Eq.(1.4). By means of the dielectric
function defined in Eq.(1.2), is possible to obtain the refractive index η (related with
the polarization), and the extinction coefficient κ of a system (related with the optical
absorption) through the equations
ε1 1
+
2
2
ε2
κ=
.
2η
η2 =
q
ε21 + ε22 ,
(1.7)
From these expressions, the main optical features of the macroscopic matter can be
derived, and constitute the starting point as input parameters for most of the standard
numerical calculations. In Fig. 1.2, we show the real and imaginary parts of the dielectric
function for noble metals comparing experimental data (Johnson and Christy [30] and
SOPRA database [31]) and from the theoretical Drude model. We can obtain a dielectric
function using a Drude model depending of a nanoparticle radius as suggest Eq.(1.4).
We present in Fig. 1.2, the case for R = 1 nm. We will address the nanoparticle size
effects in the effective dielectric function below in this manuscript. Obviously the Drude
model does a good job in describing the dielectric properties of the noble metals for the
frequencies below the onset of the interband transitions involving the d-band electrons.
1.1.2.
Plasmon resonances and nanoparticles
In systems like nanoparticles, the plasmon resonance has a surface localized nature,
known commonly as Localized Surface Plasmon Resonance (LSPR), in which the whole
electron gas oscillates with respect to the fixed lattice of positive ions in a resonant motion that depends both on geometrical and compositional parameters. A representation
of this effect is shown in Fig. 1.3. The effect of the LSPR in a small sphere or radius R,
can be explained from its frequency-dependent polarizability α(ω). In the electrostatic
limit it is given by [13]
α(ω) = 4πR3
ε(ω) − εd
.
ε(ω) + 2εd
(1.8)
where εd represents the dielectric constant of the surrounding medium. The resonant
condition occurs at localised dipole plasmon frequency ωDP when Re[ε(ωDP )] = −2εd .
At LSPR, the nanoparticle acts as an oscillating dipole with a strong absorption and
5
Chapter 1
Introduction
Figure 1.2: Real (upper figures) and imaginary (lower figures) parts of the dielectric
function of noble metals (silver, gold and copper) showing the Drude Model and
experimental data base from SOPRA database (triangles) [31] and the well known
data from Johnson and Christy (green points) [30]. The theoretical Drude model for
bulk and nanoparticles (R = 1nm and A = 0.25) corresponds to continuous lines (black
and red respectively).
Figure 1.3: Schematic illustration of a localized surface plasmon. In resonance, the
whole electron gas can oscillate respect to the fixed lattice of positive ions known as
Localized Surface Plasmon Resonance (LSPR).
scattering features. Scattering of light can be defined as the redirection of electromagnetic waves when they meet with an obstacle. The total extinction cross-section is the
amount of energy removed from the incident field due to absorption and scattering. By
energy balance, the extinction cross-section σext is a superposition of the scattering σsca
6
1.1
Plasmonics
and absorption cross-sections σabs :
σext = σabs + σsca .
(1.9)
When the obstacle is a spherical nanoparticle of radius R, the scattering σsca and absorption σabs cross-sections can be expressed in terms of the resonant polarization α,
given in the electrostatic limit by [32]
σscat
σabs
kd4 2
8π 4 6 ε − εd 2
|α| =
k R ,
=
6π
3 d
ε + 2εd ε − εd
3
,
= kd Im[α] = 4πkd R Im
ε + 2εd
(1.10)
√
where kd = ω εd /c is the wave number of the incoming light in the embedding dielectric
medium, and c is the speed of light in vacuum. Obviously the scattering and absorption
cross-section are resonantly enhanced at ω = ωDP . The effect of the plasmon resonance
is notable not only in the far field response (scattering, extinction and absorption cross
section), but in the near-field response. The electric field is enhanced in the vicinity of
the nanoparticle. In brief, these are the main features of LSPR which can be retrieved in
a rigorous retarded solutions of the electrodynamic Maxwell equations. In the left panel
of Fig. 1.4 we show the optical response of a silver sphere of radius 20 nm obtained from
the solution of Maxwell’s equations with appropriate boundary conditions [33]. The
scattering and absorption cross-section are shown in the left panel of the figure, and the
field enhancement at resonance (λ =360 nm) is shown in the right panel of the same
figure. We define the near Field Enhancement Factor (FEF or EF) as the norm of the
~ and an incident
ratio between the scattered near field of the spherical nanoparticle E
~0
z-polarized electric field E
EF =
~
|E|
.
~ 0|
|E
(1.11)
The resulting induced fields, are those of the dipolar electric resonance and show the
expected dipolar field distribution, where there is a strong local field enhancement near
the particle.
7
Chapter 1
Introduction
Figure 1.4: Optical response of a silver nanosphere with R = 20 nm. Left: Scattering
and absorption cross sections of the particle normalized to the surface area of the sphere.
Right: amplitude of the electric near-field enhancement around the nanoparticle at the
dipolar resonance (λ = 360 nm). Image taken from reference [33]. Reprinted with kind
permission of the authors.
1.1.3.
Near Field Enhacement Factor
The near field enhancement at plasmon resonance can be illustrated with solution of
~ as [34]
Maxwell equations in matter, formulated for the scattered electric field E
∇ × µr
iσ ~
2
~
∇ × E − k0 r −
E = 0,
ω0
(1.12)
where µr is the relative permeability (taken as = 1 for metallic systems), εr the complex
relative permittivity, σ is the nanoparticle conductivity related with ε(ω) through
ε(ω) − 1 = i
√
and k0 = ω ε0 µ0 =
ω
c,
4πσ
ω
(1.13)
with c, ε0 and µ0 the corresponding values of light velocity, the
permeability and permittivity in vacuum, respectively.
To solve the problem of light-nanoparticle interaction given by Eq.(1.12), we use the
frequency dependent Drude dielectric function εr (ω) as an input. The vector Maxwell
equations (in our case, only for the electromagnetic field) are then solved numerically,
by means of, for example, a standard numerical Finite Elements Method (FEM) as included in packages like COMSOL Multiphysics [34].
8
1.1
Plasmonics
Figure 1.5: Sketch of the enhancement field factor calculation. An incident electromagnetic wave generates a strong near (local) field in the nanoparticle’s vecinity. The
EFF is definded by Eq.(1.11)
A schematic picture of the system is shown in Fig. 1.5, where an incident electromagnetic
field over a nanoparticle system is shown together with a map of the field enhancement.
The dielectric function of silver is obtained via Eq.(1.7) using the empirical data for the
frequency ω- dependent optical constants η and κ as provided by Johnson and Christy
[30].
1.1.4.
Nanoparticle dimers and coupling
One of the most dramatic effects of metal nanoparticles, is their ability to exhibit extraordinary large enhancement of the near field, when they are placed in close proximity.
It is well-known that the plasmon resonances of metallic systems of nanoparticles provide
at optical frequencies, fields responsible for an electromagnetic contribution to SurfaceEnhanced Raman Scattering (SERS) [19]. When metallic nanoparticles are used as
SERS substrates, variations in SERS enhancement (often by many orders of magnitude)
have been observed. This variation has been attributed to localized plasmons formations
at the gaps between nanoparticles. Here, the interaction of the induced charges at opposite sides of the junction gives rise to the strong red-shift of the plasmon modes with
decreasing width of the gap and large field enhancements that enable SERS detection
at or near single molecule sensitivity.
9
Chapter 1
Introduction
Figure 1.6: a) Enhancement electric field factor for a dimer composed from two gold
nanospheres with R = 10 nm. The field enhancement is measured in the geometrical
middle of the structure and the light is linearly polarized along the symmetry axis that
contains the centers of the nanospheres. b) and c) present the extinction spectra and
electric field intensity (respectively) calculated for gold dimers R = 20 nm both with
(solid curves) and without (dashed curves) consideration of nonlocal effects. Different
gap distances d are considered, as labeled. In this image, b) and c) are taken from [16],
Copyright 2008, American Chemical Society.
10
1.1
Plasmonics
Figure 1.7: EF using classical electromagnetic theory in gold dimers of R = 16Rb
(left) and R = 24Rb (rigth), where Rb = 0.0529 nm is the Bohr radius. The field
enhancement is measured in the geometrical middle of the structure and the light is
linearly polarized along the symmetry axis that contains the canters of the nanospheres.
Classical theory gives a continuous increase of the electric field between particles as the
distance d decreases in opposition to quantum effects in very narrow gaps, where effects
like tunneling must be considered.
At resonance, as we said, very strong field enhancements are obtained. The way to obtain these high EF, requires to solve the full vector Maxwell equations using analytical
or numerical methods. We present in Fig. 1.6.a, the effect of the size of the plasmonic
gap d (defined in the inset) on the field enhancement in the middle of the cavity formed
by a dimer of identical gold nanospheres of radii R=10 nm. Results are obtained from
the Finite Element Method (FEM) solution of classical Maxwell equations, and are displayed as function of the wavelength of the incoming plane wave radiation. The studied
sizes of the gap between nanoparticles range from d=1 nm to d=5 nm, where the quantum effects are expected to be weak so that the classical electromagnetic theory can
be used. Observe a monotonous growth of the field enhancement at plasmon peak as
the separation d between nanoparticles decreases. Nanoparticle dimers may serve as a
simple prototype model system for the study of the important physical factors underlying the electromagnetic field enhancements [35]. From a classical point of view, this
behavior is understood due to the interference of the fields produced from the single
particle’s resonances, and as an edge effect in the cavity formed by the particles [36].
These intense and highly localized fields are known in literature as “hot spots” and
appear not only in spherical dimers, but in systems with more than two particles and
not necessarily of spherical geometry. Upon reduction of d the field enhancement can
reach several orders of magnitude as shown in Fig. 1.6.b and 1.6.c. Figures 1.6.b and
11
Chapter 1
Introduction
1.6.c show the extinction spectra and the EF in gold dimers of R = 20 nm with narrower gaps taken from [16]. The authors of this work have also considered the nonlocal
effects that can reduce plasmon couplings at fixed d and thus blue-shift the resonance
peak in the visible spectrum as compared to the results obtained with local dielectric
function of the metal. The resonance peaks in the absorption or extinction cross section
are due to the excitation of the bounding plasmon mode of the dimer and are associated with resonance field enhancement in the gap. Observe the pronounced red shift
of the resonances with decreasing d which is much stronger in the local classical theory
because of the divergent (at d → 0) coupling between plasmon modes localised at individual nanoparticles. The coupling of the resonant modes can be analytically described
using the plasmon hybridization (PH) model explained in great detail in Refs.[37, 38].
In Fig. 1.7, we show the EF obtained in this Thesis work from the classical electromagnetic theory as implemented in FEM for small gold dimers of R = 16Rb and R = 24Rb
(where Rb = 0.0529 nm is the Bohr radius). Similar to the data reported in Fig. 1.6, reduction of the size of plasmonic gap results in stronger field enhancement in the junction.
Classical electromagnetic theory may be insufficient to calculate the field enhancement
factor in sub nanometer (very narrow) gaps between particles. The tunneling phenomena
strongly affects the optical response when the gap size is near to the touching limit (below
∼ 0.5 nm in sodium dimers) [1, 39]. For the gap size d ≥ 0.5 nm, the field enhancement
increases with decreasing d, as is shown in Ref. [38, 39] (See Fig. 1.8.c and Fig. 1.8.c).
Below this separation distance, the field enhancement decreases as a consequence of
the increasing tunneling current which neutralizes the induced charges at the opposite
sides [38] of the gap. For d approaching zero (touching limit) the field enhancement is
quenched which corresponds to the continuous metal formed in the junction even prior
the direct geometrical contact between metallic surfaces [38, 39].
In Fig. 1.8 we show the electric field enhancement dependence on the gap width d and
spatial profiles of the near fields obtained in Ref. [38] for small free-electron metal dimers.
Results of the classical electromagnetic calculations are compared with full quantum time
dependent density functional (TDDFT) calculations. In this pioneering work it has been
demonstrated that the classical divergence of the near fields for vanishing gap is removed
because of the tunneling effects.
12
1.2
Quantum Plasmonics
Figure 1.8: The upper panel shows a comparison of the maximum electromagnetic
field enhancements calculated using electromagnetic calculations (red) and TDLDA
(blue) for R = 24Rb dimers of separations d = 24 (A), 8 (B), 4 (C), and 2 Rb (D).
The lower panels compare the field distributions calculated using classical model (top
panels) and TDLDA (bottom panels). Image taken from [38], Copyright 2009, American
Chemical Society.
1.2.
Quantum Plasmonics
While the optical response of the systems of tens and hundreds of nanometers is well
understood, thanks to the solution of the Maxwell equations in the classical electrodynamics framework, the engineering of nanostructures with characteristic scales at
nanometers range and below, brings quantum effects at the focus of current research in
plasmonics.
13
Chapter 1
Introduction
When electromagnetic radiation interacts with a metal junction with a width comparable
to the characteristic size of the electron density spill out from metal surfaces, the issues
of the actual position of the screening charges and of the possible electron tunneling
gains the importance.
In the classical theory the induced charges are located at the geometrical surfaces of the
metals, while quantum calculations position of these charges are of some Å inside or outside the metal surface depending on the metal. Thus, the “actual” size of the gap defined
by the separation between screening charges is different from the geometrical size of the
gap. This effect is particularly important for the range of gap widths 1 Å ≤ d ≤ 10 Å,
and it is linked with non-local dynamical screening at surfaces, phenomenon, which also
controls the shift of the frequency of the plasmon modes of individual nanobjects as
function of their size. The non-local screening can be partially taken care off within
the framework of the classical Maxwell equations using the model non-local dielectric
functions. The so-called Hydrodynamic approach is one of the most popular nowadays
because of its physical transparency and numerical efficiency [40].
For even narrower gaps (below 1 nm), the electron tunneling is the dominating quantum
effect which strongly affects the plasmon modes of hybrid nanostructures. Both far-and
near-fields can be altered as illustrated with Fig. 1.8. As we will show in this thesis, the
electron tunneling effect has a great importance as a principle of action in optoelectronics. To date, the description of this intimately quantum effect has been possible in a few
nanometer scale nanostructures, in which the full quantum treatments such as TDDFT
are feasible. These canonical systems comprise nanoparticle pairs (dimers), and coreshell systems (known in literature as nanomatryoshkas), where the quantum leap of the
electrons is controlled by the distance between the metal surfaces of the nanoparticles
forming the cavity (see Fig. 1.9). The quantum effects on LSPR have motivated the
plasmonics community to critically reexamine the description of the optical response of
small isolated nano-particles and plasmonic nanostructures with narrow gaps by revisiting the existing models from both classical and quantum points of view. In Figure 1.10,
we show several structures presenting quantum effects. Such structures are the main
systems studied in this thesis. In this kind of systems, where the quantum effects as
tunneling are explicit, the calculations of the optical response represents a real challenge.
Indeed, quantum treatments beyond classical electrodynamics are required, but these
are computationally inaccessible nowadays for the plasmonic systems of actual interest
14
1.2
Quantum Plasmonics
Figure 1.9: Nano-optics in plasmon cavities. Sketch of the interaction of linearly
polarized light with a cavity made by a plasmonic nanoparticle dimer and a nanomatryoshka. The interaction between the electric charges induced at the surfaces of the
nanostructures (in the gap region) increase the coupling between plasmon modes localised at individual nanoparticles and enhances the electric field in the gap. In subnanometric cavities, the interaction between plasmonic fields and conduction electrons
leads to a variety of physical processes that require quantum mechanical calculations.
These processes may be exploited in optoelectronic applications, as we present in the
present document.
that comprise millions of electrons. As a consequence, the ongoing theoretical effort is
devoted to the study of the main quantum effects in metallic nanostructures described
at full atomistic ab initio level or using free-electron models. Essentially, the size of the
nanostructure is small enough so that the full quantum calculations mainly based on
TDDFT can be performed. The knowledge acquired in these studies is then used in
elaboration of the (semi-classical) approaches allowing to include the quantum effects
within the framework of the Maxwell equations so that the realistic structures can be
addressed with (i) account for quantum effects, and (ii) using efficient numerical solvers
developed in classical electrodynamics [1, 2, 38, 39, 41–43]. Such theories, dealing with
quantum nature of electrons responding to the classical electromagnetic wave become
a part of a more general field research, called quantum plasmonics, that studies the
implications for plasmonics of the particle-wave duality of both: photons and electrons.
15
Chapter 1
Introduction
Figure 1.10: Different geometrical configurations of nanostructures studied in this
thesis. We studied the interaction of light with individual nanoparticles, nanoparticle
dimers, and core shell systems that are known as nanomatryoshkas.
1.2.1.
Active Quantum Plasmonics
Up to now, we have outlined the main phenomena determining the plasmon modes
of the metallic nanostructures with narrow gaps.
Currently the plasmon response
is tuned mainly during the nanostructure manufacturing by means of designing and
engineering the nanoparticles material, size, shape, and the refraction index of the
dielectric surrounding [29]. However, it would be highly desirable to dispose fast, adaptive and reversible procedures to control the plasmon response. Recent experimental
developments suggest several possibilities for such active control, for example by using
flexible substrates [44, 45], liquid crystal environments [46, 47], electrically induced
heating [48, 49], optical modulation using arrays of quantum dots [50], or excitation
of free carriers [51, 52]. This last option is the only one that provides access to electronic time scales (ps), however, it requires relatively high power lasers. In this context,
flexibility and simplicity of the electrical modulation of the optical response with applied
16
1.3
Summary
bias as achieved in 2D materials, such as graphene [53] appears extremely attractive.
Two-dimensional materials allow electrical control of their plasmon response in the
terahertz to infra-red frequency range via electron injection with applied bias. This
is possible thanks to the overall low concentration of free charge carriers in these materials. Metal nanoparticles have high density of conduction electrons that allow a plasmon
resonances in the optical range. But precisely because of this high density of conduction
electrons the control strategies based on the electron/hole injection are not efficient in
this case. However, as we will show in this thesis, quantum effects in narrow plasmonic
gaps offer a new possibility of the active control of plasmon response. In a joint effort
with theoretical groups from France2 , Spain3 , and United States4 , a concept that involves electrical control by an applied bias of the plasmon resonances of the canonical
metal-insulator-metal plasmonic structure characterized by a very narrow junction is
proposed. When the electron tunneling through the junction is possible, a DC bias
applied across the junction allows modifying the tunneling barrier and thus the conductivity of the junction. The latter in turn affects the plasmon modes of the system.
In Fig. 1.11 we show a sketch of the proposal, which will be explained in Chapter 5
Recently, Savage et. al. [54] have demonstrated how the quantum tunneling through
the narrow gap between plasmonic nanoparticles (in a dimer geometry) strongly modifies the response of the system to the external light. In Ref. [54] the conductivity of
the junction has been varied by changing its width. In analogy with scanning tunneling
microscopy, we propose to reach the sought change of the conductvity via an applied DC
bias. This is precisely the active control mechanism that we will explore and promote
in a practical way in this thesis.
1.3.
Summary
The present thesis is focused on the study of how the quantum effects change the
plasmon properties of sub-nanometric systems. In particular, we address the systems
characterised by the narrow gaps between metal nanostructures where the optical response
is strongly affected by, and can be controlled with electron tunneling effects. Using
2
Andrei. G. Borisov and Dana-Codruta Marinica. Institut des Sciences Moléculaires d’Orsay, Université Paris Sud, Paris-France
3
Javier Aizpurua and Andrei K. Kazansky, Nanophotonics Group. Materials Physics Center and
Donostia International Physics Center. Donostia-San Sebastian-Spain
4
Peter Nordlander. Rice University, Houston-USA
17
Chapter 1
Introduction
Figure 1.11: Schematics of the mechanism underlying the proposed bias-control strategy for active plasmonics.
several approaches, from classical electrodynamics and analytical quantum mechanics
towards more sophisticate models like quantum corrected model, and full quantum
TDDFT, allows us to cover different topics of current interest in the field. Our research
is outlined in this thesis manuscript which is organised as following:
Starting with the present chapter (Chapter 1) we have shown some generalities,
state of the art and the main topics in the field of (quantum) plasmonics.
In second chapter (Chapter 2), we present the study of the quantum size effects in
the optical response of individual metallic nanoparticles of spherical geometry, and
using different types of confinement potential. Firstly, we present the improvement
of the classical models used up to now. We use electronic states in the confinement
potential to derive the dielectric function and apply this dielectric function to study
optical response of the corresponding metal nanoparticle. Thus, the limits of the
different regimes can be defined. In particular, we show that particle sizes smaller
18
1.3
Summary
than 10 nm are of special interest due to their natural quantum features that make
them an interesting subject to understand the limit of pure classical phenomena.
In the third (Chapter 3) and fourth (Chapter 4) chapters, we extend different
previous studies on spherical nanoparticles (or clusters) and cylindrical nanowire
dimers, where the tunneling current occurs in extremely localized gap regions. We
perform the quantum mechanical time dependent density functional theory calculations of the plasmonic response of cylindrical core-shell nanoparticles (nanomatryoshkas). We show that the full quantum results can be recovered in the classical
calculations using the Quantum Corrected Model (QCM). Additionally, using the
QCM we retrieve the quantum results for the absorption cross section in spherical
nanomatryoshkas as obtained earlier [3]. Thus, our results establish the applicability of the QCM for a wide range of geometries that hold tunneling gaps[4]. We
present the Quantum Corrected Model (QCM) as an alternative for describing
the electron tunneling in very small gaps in hybrid nanosystems like the so called
metallic nanomatryoshkas (NM).
In Chapter 5, we propose a mechanism for fast and active control of the optical
response of metallic nanostructures based on exploiting the quantum regime in
subnanometric plasmonic gaps. The sough active control is reached by applying
an external DC bias across a narrow gap in core shell nanostructures (cylindrical
nanomatryoshkas). We demonstrate the feasibility of the concept of active control
using the TDDFT calculations of the optical response of this kind of structures in
presence of an applied DC bias. We show that the optical response of a metallic
cavity can indeed be electrically controlled thanks to the bias induced modification
of the tunneling barrier separating metallic surfaces across the gap.
Finally, in Chapter 6, we use a TDDFT to perform a full quantum calculation
of the optical response, and electron loss for charged spherical clusters. We have
obtained that, because the system stays neutral in the bulk, and the extra-charge is
localized at the surface of the cluster, the dipolar plasmon mode displays only small
frequency shift linked with change of the electron spill-out from the nanoparticle
boundaries. In the case of the negatively charged clusters, even small (relative to
the total number of electrons in the system) negative charge raises Fermi level of the
cluster above the vacuum level and renders the system unstable. The extra charge
of the cluster decays with characteristic time constants that can be fully understood
using an analytical study based on the WKB method, similar to the studies of the
alpha-decay. Our results on charged clusters allow critical reinterpretation of the
19
Chapter 1
Introduction
experimental data obtained with electrochemistry, where the shift of the plasmon
resonances have been attributed to the charging effects. Based on our results, we
tentatively propose that the change of the dielectric environment in immediate
vicinity of the cluster is the origin of the shift of the plasmon resonance.
We will close the thesis with some general conclusions and summary of the main
original results in Chapter 7.
20
Chapter 2
Confinement potential effects on
the Localized Surface Plasmon
Resonance of metallic small
nanoparticles
In this chapter, we analyze the importance of a description of the electronic transitions
in small nanoparticles through the optical response to small changes of size in systems
whose dimensions are in the subnanometric scale. We present the calculations of the
optical response of that kind of systems by using the exact eigen-energies and wave
functions for nanospheres with diameters smaller than 10 nm, to obtain the dielectric
function under different conditions of confinement. The main aim is to use the so obtained dielectric function to calculate the absorption spectra of one electron confined
in a sphere in two cases: i) an infinite spherical confinement and ii) a finite spherical
confinement, in which, the value of the wells depth is calculated after adjusting the
number of atoms that composed each sphere, so that the energies and dipole matrix
elements give a more accurate information of the optical response. Moreover, we extend
the calculation of this dielectric function for obtaining the optical constants needed to
find the plasmon frequency, through a numerical finite element method (FEM) solving
the Maxwell’s equations, in order to obtain the enhancement of the near electric field.
We show an interesting behavior for particles sizes less than 10 nm, finding that the variation induced in the eigen-energies, through slight size changes in the particle, provides
significant variations in the optical response of these nanoparticles. This effect, can be
21
Chapter 2 Confinement potential effects on the Localized Surface Plasmon
observed in the optical absorption spectra and in the localized surface plasmon energies as confirmed in recent observation of plasmonic phenomena at the subnanometer to
atomic scale [55].
2.1.
Introduction
The localized surface plasmons (LSP) in nanoparticles are subject of analysis because
they enable strong optical absorption and scattering in subwavelength structures. These
properties depend on the nanoparticles material, size, shape, and the refraction index of
the surrounding material [29]. Each one of these parameters is worth of being studied
in order to understand the optical behavior at the nano-scale. Most of the calculations assume the Mie model [11] because it shows good agreement with experiments in
nanoparticles up to 10nm size using optical constants of Johnson and Christy [30]. For
smaller particles, this model achieves poor agreement, and in particular, when dealing
with particles in the size-range of 1 − 10nm, the surface becomes more and more important compared to the bulk response and thus influences heavily the collective modes
localized at the surface region. On the other hand, the quantum effects should be significant due to the electrons could be able to spill out of the particle under finite confinement,
and new interesting effects appear for these systems that are in the subnanometric sizes.
Plasmon oscillations and electron spill out could balance tuning the optical response
of the particles, as proposed recently Monreal et al, [56] where they examine electron
spill out effects, which can induce red or blue shifts of the surface plasmon energy, that
can be comparable to or even stronger than the quantum size. However, their model
remains classical in spite of introducing a semiconductor like gap at the Fermi level.
However, for subnanometric sizes a quantum description needs to be considered. Following this idea, we use as a starting point the question of quantization, showing an
exact calculation of the dielectric function, compared with other approaches and with
accurate experiments, in order to examine the optical properties of systems in the limit
of subnanometric size scale. In fact, we focus this chapter on the quantum description
of optical response of ultra-small spherical nanoparticles, whose diameters are less than
10 nm. We describe the dielectric function of these systems starting from the assumption that one electron confined in the nanoparticle only can have discretized energies
and we make carefully calculations of the allowed electron energies and the corresponding wave-functions for infinite and finite confinement cases as function of the particle
size (diameter). In both situations we use the analytical complete eigen-functions to
22
2.2
obtain the dielectric function, which give us confidence in the description of the optical
response. We test our results of the dielectric function through optical absorption and
enhancement field factor, which allow us to compare with recent experimental results
[55]. Starting from an infinite spherical confinement with asymptotic values for energies
and wave functions (Model A) and exact calculation of both energies and wave functions (Model B), we examine the effect of the exact values on the dielectric function
for ultra-small particles. These results are also compared with finite spherical wells confinement, whose depth is obtained from the number of atoms in each particle (Model C).
In his chapter, we describe the optical absorption of one particle confined in an spherical
well, where we can compare our results with experiments, and discuss the differences
in absorption coming only from quantum confinement, when the electrons have to stay
within the particle, and when they have a probability of being outside depending on the
particle size. We also show a reproducible behavior of the plasmon energy depending
on confinement, and discuss our main results comparing them with experiments and
theoretical models based on the spill out effect.
2.2.
When the size of a nanoparticle becomes smaller than 10 nm, the continuous electronic
band of a nanoparticle breaks up into observable discrete states. Such effects have been
observed in experiments with metallic nano-particles, [26, 55] showing a notably difference between the results obtained in the 10 − 100 nm regime and the ‘quantum regime’
(particles below 10 nm of radius).
As a first approximation, to describe the optical response of an electron gas in a metal
nanoparticle, Genzel and Martin [26] showed a simplified quantum model where the free
electron gas is confined in an infinite cubic potential well (see Appendix. A), where the
dielectric function of a single metal nanoparticle, under the influence of an electromagnetic wave with frequency ω and z-polarization, is given by
ωp2 X sif (Fi − Ff )
ε(ω) = ε∞ +
2 − ω 2 − iωγ ,
N
ωif
if
i,f
23
(2.1)
with the terms ε∞ , sif , ωif and γif are respectively, the interband contribution of
core electrons, the oscillator strength, the frequencies and the damping for the dipole
transition from an initial state i to another final state f ; and Fi and Ff are the values
of the Fermi-Dirac distribution function for the initial and final states. The oscillator
strength sif corresponds to the dipolar form in the z-direction in terms of the initial |ii
and final |f i states
2m∗ ωif
|hf |z|ii|2 .
~
sif =
(2.2)
where the parameter m∗ = me /N is given in terms of the mass of electron me and the
total number of confined atoms N . Using different types of confinements, we calculate
the eigen functions and energies, within the A, B and C models described before [57].
2.2.1.
Free electron gas in a spherical infinite well
confinement: Exact solution
We consider an infinite spherical potential well to solve the Schrödinger equation for one
electron, with the appropriate boundary conditions finding a set of wave functions of
the form [58]
1
ψ(r, θ, φ) =
|jl+1 (αnl )|
r
2 αnl m
jl
r Yl (θ, φ),
R3
R
(2.3)
where jl represents the spherical Bessel functions, Ylm the standard spherical harmonics,
αnl is the n-th zero of jl and R the radius of a spherical particle. The eigen-energies
En,l of this problem in a quantized form associated to the l-th spherical Bessel function
are given by
En,l =
2
~2 αln
.
2m∗ R2
(2.4)
The oscillator strenght in Eq.(2.2) can be calculated, taking z = r cos θ in the expression
Z
2π
Z
π
Z
|hf |z|ii| =
0
×
0
R
drdθdφr3 sin θ cos θ
(2.5)
0
Ψ∗nf ,lf ,mf (r, θ, φ)Ψni ,li ,mi (r, θ, φ),
24
2.2
(li + mi + 1)(li − mi + 1)
δ∆l,+1
(2li + 1)(2li + 3)
s
(li + mi )(li − mi )
+
δ∆l,−1
(2li + 1)(2li − 1)
(2.6)
where the angular integral has the form
s
Iang =
and, the radial term,
Irad
1
1
=
|jlf +1 (αnf lf )| |jli +1 (αni li )|
2
R3
Z
×
R
drjlf
αn
0
α
ni li
r r3 jli
r .
R
R
f lf
(2.7)
Then, the integral to calculate is simply
|hf |z|ii| = Iang × Irad
(2.8)
which is different from zero only for values ∆l = lf − li = ±1. This description corresponds to our labeled Model B.
2.2.2.
Free electron gas in a spherical infinite well
confinement: Approximate solution
One way to obtain a simplified solution of the problem is to use the approximation [55]
~2 π 2
(2n + l + 2)2
8M R2
2
~2
l
=
π n+ +1
,
2M R2
2
En,l =
(2.9)
for the eigen-energies, and the asymptotic form for the wavefunctions [58],
jl (x) ≈
h
i
1
π
cos x − (l + 1) ,
x
2
(2.10)
valid only for x l2 /2 + l.
Eq.(2.4) (Model B) and Eq.(2.9) (Model A) give us different values for nanoparticles,
for example with R = 1nm. We can observe a good agreement only for l = 0, and a
significant overestimation for l > 0. Besides this, in Fig. 2.1 we show jl (kr) = jl (αln r/R)
and its asymptotic approximation given by Eq.(2.10) versus r/R and n = 1. As we can
25
Figure 2.1: Comparison between jl and its asymptotic approximation (2.10) versus
r/R for different l’s and n = 1.
see, the approximate and the exact Bessel functions are identical for l = 0, but they
differ remarkably for the l = 1, 2, and 3 quantum numbers. Therefore, we conclude that
the use of exact wave functions (Model A) are absolutely needed to give an accurate
description of the optical response in these systems.
2.2.3.
Quantum finite confinement
Going a step further, we consider a finite spherical confinement, which is related with the
number of atoms contained in a spherical nanoparticle. We fit a relation derived of ab
initio calculations made by He et. al. [59], by taking the number of particles as function
of its radius R. Based on this relation, then we calculate the last occupied electronic
state following the Pauli exclusion principle. This energy value is defined as the Fermi
level and varies with the nanoparticle radius. To establish the depth of the well, we
consider the work function W (R) for small metallic nanoparticles following Ref. [60],
which is defined as the energy required to remove one electron from the nanoparticle,
which is given by
W (R) = 4.37 +
26
5.4
,
R(Å)
(2.11)
2.2
Figure 2.2: Schematic diagram illustrating quantum infinite potential and quantum
finite potential, to evaluate the dipolar contributions in the dielectric function. Here
we show some transitions when the energy levels are quantized.
where the 4.37 value is the average work function for bulk Ag (in eV) and 5.4 = 38 e2 is a
parameter that accounts for the difference in the work function for a conducting plane
and a sphere [60]. Considering this value, we define the depth well DW as the sum of
the Fermi energy level EF and W (R) for each nanoparticle radius, which reads as
DW (R) = −(EF (R) + W (R)).
(2.12)
Using this result, we solve the Schrödinger equation numerically, calculating the eigenvalues for each nanoparticle. In order to obtain the corresponding dielectric function,
we need the oscillator strength for the allowed transitions using the wave functions of
infinite confinement, which are needed to obtain the dielectric function as can be seen in
Eq.(2.2). We choose transitions between occupied states just below the Fermi level and
unoccupied states above the Fermi level, and transitions between the unoccupied state
just above the Fermi level and occupied states below the Fermi level, in a wide range
(0 − 6eV), following the selection rule ∆l = ±1, which gives us a complete information
of the allowed electronic transitions in our systems. We show in Fig. 2.2, a schematic
diagram that illustrates this assumptions for quantum B and C Models.
27
2.2.4.
Confinement effects on the optical response
Now, we calculate the dielectric function based on Eq.(2.1), and present the imaginary
part of the dielectric function for several nanoparticle sizes in Fig. 2.3. The continuous
black lines, corresponding to the Model A, show besides the main peak, several possible
transition with smaller intensity, while the dashed lines, Model B, reduces significantly
the number of transitions in the region of interest, as an effect of the oscillator strenght.
On the other hand, the energy position of the main peaks is always blue-shifted with
respect of Model B for each radius.
With the aim of examining the trend of the peaks of energy in the imaginary part of
the dielectric function as a function of radius and to compare the results of the three
models we present Fig. 2.4. While model A predicts a monotone blue-shift as the particle
radius decreases, whose value reaches almost 2 eV for R = 1 nm, Models B and C, show
a maximal blue-shift about 1 eV. However, the shifts between consecutive diameters
oscillate.
2.3.
Absorption spectra
In this section, we compare the absorption peaks following the three approaches described
above. To calculate the absorption spectra, we use the expression
3/2
9f ωεm Im[ε(ω)]
,
σext (ω) =
c(Re[ε(ω)] + 2εm )2 + (Im[ε(ω)])2
(2.13)
where f is a fraction of volume of the sphere in the media, εm corresponds to the
dielectric constant of the surrounding media and c the light speed [61]. For particles of
radius 5nm there is no appreciable difference between the three calculations (see Fig.
2.5), showing all of them a peak at 3.3 eV, while for smaller particles we obtain both
red and blue shifts that are coming from the confinement type. Looking from the top
to the bottom of Fig. 2.5, the particle radius decreases and Model A (at the left), show
a rather smooth blue-shift while Models B and C (center and right), present red and
blue shifts alterning as function of the nanoparticle radius.
We observe oscillations
of the maximal absorption for nanoparticles smaller than 10nm (R = 5nm). The high
dispersion of the experimental EELS measurements [55] in this region also shows big
28
2.3
Absorption spectra
Figure 2.3: Imaginary part of the dielectric function for a silver nanoparticle
as function of radius based on Model A (continuous lines) and Model B (dashed
lines).
oscillations for particles smaller than 10 nm of diameter. Our predictions are located below the region of experimental error, which can be corrected through the ε∞ parameter.
The three approximations are in good agreement to each other and to the experiment,
showing changes of red and blue shifts at the same particle sizes.
With the aim of testing our three models of dielectric function related to the localized
surface plasmon resonance in silver ultra small nanoparticles, we present in the next
section the enhancement field factor.
29
Figure 2.4: Main peak in the imaginary part of the dielectric function for each
model used. Squares represent the Model A, black points Model B and triangles
Model C.
Figure 2.5: Absorption spectra for several particle radius, using Model A (left),
Model B (center) and Model C (right).
30
2.4
2.4.
Enhancement of the near field
We calculate expressions for the dielectric function depending on the incident electromagnetic wave frequency εr (ω) as input parameters, which are calculated for each particle
size and confinement potential, by solving the vector Maxwell equations numerically, by
means of a standard finite elements method (FEM) using the COMSOL Multiphysics
package [34], as we defined in the last chapter.
The obtained resonances give the frequencies of the surface local plasmon, and we observe that we obtain values located at the same energies as the absorption peaks, which
follow the same behavior as discussed earlier.
If we look at the field factor for each particle radius in Fig. 2.6 from bottom to top,
we observe a red and blue shifts for all particle radius as a whole, depending on the
used model. Comparing Model A with Model B, the change is due to the dipole matrix
elements (oscillator strength), while the differences between Model A and Model C are
due to transition energies. Both effects show lower plasmon energies than the approximate Model A, maintaining the same parameters in the three models. However, they
show an increase in the enhancement field factor of about 4 times (compared with model
A). The behavior of the plasmon frequency within each model, shows oscillations, that
means red and blue shifts alternating. Finally we present in Fig. 2.7, another feature
that is interesting comparing with EELS experiment, which is the energy width of the
region where the oscillations are detected. In Model A, the width achieves more than
1.5 eV, while for model B and C is only of 1 eV. On the other side, the experimental
measurements show values ranging from 0.8 eV, with maximal error bars for 5 and 3 nm
diameters, where small changes in the particle size can give strong shifts, which depend
on the sensitivity to the well depth. It means to the eigen energies and corresponding
oscillator strengths. As the particle decreases from 10 nm, the enhancement field factor decreases, indicating a weaker near field because the number of electrons that can
resonate are getting less and less. However, for this range of nanoparticles the ratio
surface/volume increases considerably, giving rise to strong dependence on the energy
differences and oscillator strength between them.
In Fig. 2.7, we show the localized surface plasmon resonance frequency, obtained by
the maximum of enhancement field factor for each particle size and we observe that
for particles smaller than 10 nm our three models provide the same behavior, showing
31
Figure 2.6: Enhancement Field Factor for a silver nanoparticle as function
of radius based on quantum approximate model (model A) [55], infinite exact
model (model B) and finite confinement (model C) of one electron in a spherical
particle.
alternating red and blue shifts as the particles size decreases. In the inset we present
the experimental EELS reported for the plasmon frequency in the same sizes range
obtained by Scholl et al [55], in which we can observe error bars of about one half
electron volts in the energy plasmon spectrum for nanoparticles about 5nm and 3nm.
However, our three models give plasmon energy values for several particle sizes, showing
in this spectrum, a kind of oscillations, as the particle size decreases. The confinement
effect is only important for very small nanoparticles, as expected. In a recent paper
[56] is discussed the screening effect on the surface plasmon compared to quantum size
effect for ultra small nanoparticles, less than 10nm. They show a monotone behavior, as
32
2.4
Figure 2.7: Plasmon energy as a function of the spherical nanoparticles diam-
eter. Squares, dots and triangles show the results when Model A, B and C are
used. The inset present the experimental EELS results obtained by Scholl et.
al. in [55].
result of a balance between size effect (blue shift) and spill out effect (red shift) based on
pure classical calculation. However, when comparing with EELS experiments, the big
error bars in the region of interest screen detailed optical response in this region. Our
claim is that there are clear oscillations within the error bars that can be understood as
confinements effects due to the energy differences and their allowed transitions, which
are extremely sensitive to small changes in the nanoparticle size. Therefore, we conclude
that a more sophisticate model that can account for a collective description of the optical
response is needed.
33
2.5.
Summary
We have examined using different analytical and numerical approaches, the optical
response of ultra-small metal nanoparticles, varying their sizes from 10 to 2 nm diameter
in steps of 0.5 nm in order to observe the sensitivity to small particle size changes. The
reason for the oscillations in optical response is that very small changes in the energy
spectrum induce significant and uncontrollable changes in the allowed electronic transitions. We also compare two types of confinement, using two sets of wave functions
and we conclude that red shifts in the plasmon energies are mainly due to the oscillator strengths, while the eigen-energies for finite and infinite confinement taken from
the exact zeros of the Bessel functions (Models B and C) do not strongly influence the
plasmon energies. But both of them differ considerably from the calculation based on
asymptotic values (Model A).
We have shown that exact quantum calculations, based on optical response of one electron confined in a sphere, open a way to understand the optical properties of ultra-small
nanoparticles, which is to be completed with many body effects that surely appear in
this region of particle sizes that limits the nanoscale and enters in the atomic scale.
In the region beyond nanometric scale, we could find optical response of nanoparticles
strongly dependent on the change of the size due the changes of the allowed electronic
transitions, within the discrete energy levels that should be taken into account carefully
to understand the plasmon behavior in ultra small nanoparticles that are composed of
hundreds of atoms.
Other effects that should be taken into account to understand the optical response of
ultra-small nanoparticles, undoubtedly require microscopic accurate description such ab
initio calculations, in order to improve the models presented in this chapter. We need a
more empirical effective theory that allows us to describe the phenomena observed experimentally paying the price of a less detailed description of the effects of local response
in the optical quantum subnanometric regime. The models presented in this chapter,
can be consider as a first appoximative models, and have not the formalism of selfconsistent calculations, typically used in this kind of systems. Nevertheless, is a good
starting point to understand more suitable and complex calculations, as we will show in
the next chapter.
34
Chapter 3
Quantum effects in the optical
response of core-shell
nanostructures
Quantum effects as nonlocal screening and electron tunneling can strongly affect the
plasmonic response of hybrid nanostructures. Number of recent works demonstrated on
the example of several systems that these quantum mechanical effects can be taken into
account semi-quantitatively within a classical framework of Maxwell equations. Thus,
the so-called Quantum Corrected Model (QCM) allows to account for electron tunneling
in narrow gaps between metal nanoparticles. In this chapter, we extend previous studies
on spherical clusters and cylindrical nanowire dimers where tunneling and hybridization effects occur in extremely localized regions near the touching point. We perform
quantum mechanical TDDFT calculations of the plasmon response of the axially symmetric cylindrical core-shell nanoparticles known as nanomatryoshkas (NM), where the
tunneling region extends over entire core-shell gap. We benefit from this axially symmetric situation to address relatively large (for full quantum simulations) systems of up to
12 nm in diameter. We have obtained that for core-shell separations below 0.5 nm, the
standard classical calculations fail to describe the plasmonic response of the cylindrical
nanomatryoshkas, while the QCM can reproduce quantum results. Using the QCM, we
also retrieve the quantum results for the optical response of spherical nanomatryoshkas
(as previously calculated by V. Kulkarni et. al. [3]). The comparison between the model
and the full quantum calculations establishes the applicability of the QCM for a wider
35
Chapter 3
Quantum effects in the optical response of core-shell nanostructures
range of geometries that hold tunneling gaps.1
3.1.
Introduction
Modern nanotechnology allows for fabrication of metallic nanoparticles and nanoparticle assemblies of different geometry, structure, and composition [8, 54, 55, 62–71]. This
extraordinary opportunity, in turn, opens a possibility to engineer the plasmonic modes
of these artificial nanostructures and thus the way they interact with light [72–76]. In
this context much of the interest has been devoted to the nanostructures that form
narrow plasmonic gaps between their constituent metallic surfaces [73]. The strong interaction of the plasmon-induced charge densities across the gap results in the manifold
enhancement of the confined fields and in the hybridization of the plasmonic modes
of the individual nanoparticles [77–79]. The field enhancement and the geometrical
tunability provided by plasmon resonances find many practical applications including
sensing [8, 80–83], plasmon rulers [62, 73, 84–88], and non linear optics [23, 68, 69, 89–
94], among others. So far, the classical electrodynamics framework based on the local
description of the metal dielectric function appeared adequate to describe optical properties of strongly coupled plasmonic nanoparticles separated by narrow gaps. However,
recent experimental [54, 55, 64, 69, 95–97] and theoretical [1, 38, 39, 41–43, 54, 98–100]
studies have demonstrated the importance of quantum mechanical effects when particleto-particle separations are reduced below the nanometer. In this case, the non-local
screening and electron tunneling across the gap can reduce (and even quench) the field
enhancement, and alter the optical response of the system. Because of the tunneling
a conductive contact between the nanoparticles can be established prior to the direct
geometrical overlap. As a consequence, the extinction resonances arising from the hybridization of the plasmonic modes of the individual nanoparticles disappear from the
spectrum. At the same time, a set of charge transfer plasmon modes emerge as a consequence of the electron flow between the nanoparticles [98, 101–105]
1
The graphs and the text including in this chapter, correspond mainly to the work published in
Optics Express referenced in [4]. BEM codes were devoloped by Javier Aizpurua from the Theory
of Nanophotonics Group in the Materials Physics Center CFM, Donostia-San Sebastian, Spain and
Donostia International Physics Center (DIPC). We adapted them for our purposes with the valuable
help of Rubén Esteban from the same affiliation. TDDFT codes used in this chapter, where shared by
Andrei Borisov from Institut des Sciences Molèculaires d’Orsay (ISMO), Universite Paris Sud, France,
and adapted for our purposes.
36
3.1
Introduction
The use of nonlocal dielectric functions [41, 106–109] captures part of the quantum behavior of such systems introducing a smooth variation of the screening electron density
profile at the interface between materials in contrast to the classical theories which admit infinitely sharp spatial variations of the screening charge densities. This solves the
problem of the divergence of the electric fields in the junction and of the energy shifts
of the plasmon resonances with decreasing junction width, as obtained in local classical
theories [110]. The latest developments of the non-local hydrodynamic descriptions allow to introduce the realistic electron density profile at the surface so that full quantum
results can be retrieved for individual nanoparticles [111] albeit at the price of growing
the numerical complexity. However, to account for tunneling across narrow interparticle junctions requires a special treatment that goes beyond classical local or non-local
hydrodynamic approaches. Quantum tunneling thus imposes a real challenge to any
theoretical description. Indeed, the simplest strategy to tackle the effect of tunneling
would consist in performing full quantum calculations of the plasmonic response, as recently reported, within an atomistic ab initio or free electron description of the metal
nanoparticles [1, 38, 39, 41–43, 54, 99]. However, because of numerical constraints,
quantum calculations can only address systems which are much smaller than those of
practical interest in plasmonics. A possible solution to this constraint consists in model
local dielectric functions that account for the quantum tunneling, in a way similar to the
derivation of the macroscopic permittivity of a system from the microscopic quantum
polarizabilities of the constituent atoms or molecules [112]. This is achieved with the
Quantum Corrected Model (QCM) [1, 2] that treats the junction between the nanoparticles as an effective medium mimicking quantum tunneling within the classical local
dielectric theory. So far, the QCM has been shown to correctly reproduce the full quantum results in plasmonic dimer structures with a localized contact region supporting the
tunneling current [42, 54, 95, 97, 100].
In this chapter, we study a system formed by a cylindrical metallic core and a cylindrical metallic shell, separated by an extended tunneling contact region between the
core and shell metal surfaces, so-called cylindrical nanomatryoshka (CNM). These kind
of hybrid nanostructures present a large tunability in their plasmonic response which
considerably explains their practical and fundamental interest. Provided current fabrication techniques NM-like structures of small size can be produced with very narrow
gaps between the core and the shell [70, 71]. Several studies addressed optical properties
of NMs within classical electromagnetic theory framework which appeared sufficient in
explaining the available experimental data [113–121]. It is only recently that the full
37
Chapter 3
quantum results have been reported for the spherical NM [3] showing the importance of
tunneling between the core and the shell.
In this chapter we address yet unexplored case of quantum regime in cylindrical NM, with
particular emphasis at the comparison between the full quantum TDDFT results and
results obtained using the QCM within the framework of classical Maxwell equations.
We study in great detail the evolution of the plasmonic resonances of the cylindrical NM
upon variation of the size of the gap between the core and the shell. We focus on the
quantum effects in this system where the standard classical calculations fail to describe
the plasmonic response of the cylindrical NM for core-shell separations below 0.5 nm,
while the QCM can reproduce well the quantum results. We also demonstrate that
the QCM reproduces the quantum results obtained previously for the spherical NM [3],
offering an efficient way to address tunneling effects in core-shell nanoparticles. Thus,
together with previous studies of plasmonic dimer structures with tunneling regions
localized around the contact point, our results for NMs demonstrate the validity of the
QCM for broad range of geometries of the tunneling regions. Indeed, in the present case
the tunneling current flows transversally through the entire core-shell gap.
3.2.
System and computational aspects
In Fig. 3.1, we show the sketch of the cylindrical and spherical core-shell nanostructures.
We have taken the name for these structures from the real russian dolls (or matryoshkas),
which gives an accurate name for this hybrid nanostructures. To define the systems, we
firstly start with the case of cylindrical NM where the infinite metallic cylindrical core
and metallic cylindrical shell are coaxial with the geometry set by the radius of the
core R1 , the internal radius of the shell R2 , and the external radius of the shell, R3 .
Following this notation, we will adopt the set (R1 , R2 , R3 ) to characterize the NM structure [3]. We consider the case where an incident light is polarized perpendicular to the
symmetry axis of the system. To focus the discussion on the role of the tunneling effect
in the plasmonic response, and to facilitate the comparison with earlier published work
on spherical nanomatryoshkas, without loss of generality the region between the core
and the shell is chosen to be vacuum. The results presented here, are obtained for a
fixed R2 = 90 a0 (48 Å), and R3 = 115 a0 (61 Å), while the radius of the core R1 is
varied (here a0 =0.53 Å stands for the Bohr radius). The range of considered R1 corresponds to the progressive reduction of the width of the vacuum gap between the core
and the shell S = R2 − R1 from 20 Å down to the touching geometry (S = 0). This
38
3.2
Figure 3.1: Sketch of the geometry of the nanomatrioshka. Upper: Real nanomatryoshkas or russian dolls, which inspires the names of the structures that we study
in this thesis. Lower: cylindrical (left) and spherical (right) nanomatryoshkas. In the
case of the cylindrical NM, the coaxial cylindrical core and cylindrical shell are infinite
along the symmetry axis of the system. The core has the radius R1 ; the internal radius
of the shell is R2 and the external radius of the shell is R3 for both systems. The core
and the shell are separated by a vacuum gap. The incident field with amplitude E0 ,
and angular frequency ω is polarized perpendicular to the symmetry axis of the NM.
allows an analysis of the emergence of the tunneling across the gap, and of its role in
the evolution of the NM plasmonic modes into those of the uniform cylindrical nanowire.
In our quantum calculations the metallic core and the shell are described within the
cylindrical jellium model (JM) (see for example ref. [42, 100]). The valence electrons
are bound by the uniform positive background charge representing the ionic cores. This
−1
background charge density is given by n0 = 4π rs3 /3 , where rs is the Wigner-Seitz
radius. R1 defines the position of the boundary of the positive background charge, the
jellium edge, of the core. R2 , and R3 define inner and outer jellium edges of the shell
respectively. Despite its simplicity, the JM correctly captures the collective plasmonic
behavior of the conduction electrons, and it has demonstrated a good predictive power
in the description of quantum effects in nanoparticle dimers, as follows from the comparison with experiments [54, 55], and with advanced full atomistic ab initio calculations
39
Chapter 3
[122]. We use the Wigner-Seitz radius rs equal to 4 a0 (2.12 Å) corresponding to Na
metal. In this case the JM performs particularly well in the description of the optical
pulse interaction with nanosized objects [123–126]. For noble metals, such as silver and
gold, the contribution of the localized d-electrons to the screening [127, 128] imposes introduction of the polarizable background [38] which would complicate the interpretation
of the results, and obscure the comparison with the classical Drude and QCM model
calculations. The qualitative conclusions in this chapter are robust and independent of
the particular choice of the density parameter.
3.2.1.
Optical response in the TDDFT framework
The quantum calculations of the absorption cross-section are based on the Kohn-Sham
(KS) scheme of the density functional theory (DFT) [129]. We use the adiabatic local density approximation with the exchange-correlation functional of Gunnarson and
Lundqvist [130]. A detailed description of the numerical technique can be found in
Ref. [39] and in Ref. [100]. First, the ground state electron density and Kohn-Sham
orbitals are obtained in standard static DFT calculations. Using the axial symmetry
allows to address the system with up to 290 electrons per 1 Å length. The structures are characterized by an overall work function of 2.9 eV, where the exact value
depends on the nanostructure geometry defined by the (R1 , R2 , R3 ) set. At a second
stage, the ω-dependent absorption cross-section per unit length, σabs (ω), is calculated
from the electron density dynamics induced by an impulsive perturbation, within the
time-dependent density functional theory (TDDFT) approach as
σabs (ω) =
4πω
Im[α(ω)],
c
(3.1)
where α(ω) is the dipolar polarizability per unit length of the system. Because of the
small transverse size of the system, retardation effects are neglected for the present choice
of the polarization of the incident electromagnetic wave. Consistent with TDDFT, the
classical electromagnetic calculations of the absorption cross section have been performed
within the quasi-static approximation [120, 121] using the local classical and the QCM
approaches.
The dielectric function of the metal is described within the Drude model (see Eq.(1.2))
described in Chapter 1. We use the parameters ωp = 5.68 eV and γ = 0.218 eV as
40
3.2
obtained from the fit of the classical results to the TDDFT calculations for the isolated cylindrical nanowire with radius R = 115 a0 (6.1 nm). Such nanowire corresponds
to the system with a fully closed gap between the core and the shell. The value of
ωp = 5.68 eV is lower than the nominal plasma frequency for the bulk Na given by
p
4πn0 /m = 5.89 eV, with me , the electron mass, reflecting the red shift of the dipolar
p
plasmon from the classical Mie value given, for the metallic cylinder, by 2πn0 /m.
This red shift is the finite size effect resulting from the spill out of electron density outside metal boundaries, as has been throughly studied in the context of surface physics
[123, 128, 131–135].
3.2.2.
The quantum corrected model in core-shell nanomatryoshkas
In order to analytically explain the tunneling effects in the extended small gaps of the
cylindrical nanomatryoshkas, we use the well known Quantun Corrected Model (QCM),
widely explained in [1, 2] for plasmonic dimers. Effects like quantum tunelling are not
explained using classical electrodynamics and fails to describe the coupling across subnanometer gaps, where quantum effects become important. QCM approach, models
the junction between adjacent nanoparticles by means of a local dielectric response
that includes electron tunnelling and tunnelling resistivity at the gap and describes
the electron tunneling between the core and the shell by filling the core-shell gap with
an effective dielectric medium described by a Drude model that depends on the gap
separation distance S, similar to that in Eq. 1.2, as:
ωp2
.
εeff (S, ω) = ε∞ −
ω(ω + iγeff (S))
(3.2)
The effective damping γeff models the transition from a resistive to a conductive nature
of the junction as a function of the gap separation S. Consistently with the dependence
of the electron tunneling probability on the size of the gap, γeff is given by
γeff = γ0 exp [S/∆] .
(3.3)
For the tunneling contact between Na metal surfaces we use ε∞ = 1 in Eq. 3.2 and
in Eq.(3.3) we use γ0 = 0.218 eV and ∆ = 0.75 Å, which are taken from references
[1, 2], and are related to quantum-mechanical calculations of the electron transmission
probability through the potential barrier separating the two metals. For sufficiently large
41
Chapter 3
Figure 3.2: (a) Absorption cross section per unit length. The green line shows the
TDDFT result for an individual nanowire with radius R = 61 Å. The rest of the results
are obtained using classical electromagnetic theory within a nonretarded approximation.
Black dashed line: individual nanowire with radius R = 61 Å. Blue line: cylindrical
shell with internal radius R2 = 47.7 Å, and external radius R3 = 61 Å. Red line:
cylindrical (37.1, 47.7, 61) Å nanomatrioshka. Insets show the direction of the radial
−
+
electric fields associated to the ω−
, ω−
, and ωc+ modes of the core-shell structure.
The arrows indicate the corresponding absorption resonances. (b) Schematic charge
distribution for the different plasmonic modes identified in a).
separation distances S → ∞, the effective damping γeff → ∞, and the QCM becomes
exactly equivalent to the local classical approach. In this situation, no tunneling is
possible and the vacuum gap limit is retrieved with εeff (S, ω) = 1. For S → 0 the
junction becomes metallic with permittivity equivalent to that of the bulk metal so that
the NM responds as a homogeneous metallic cylinder.
3.3.
Plasmon modes of a nanomatryoshka
To characterize the optical response of the cylindrical NM with particular emphasis on
the assignment of the plasmonic modes, we base our discussion on the classical quasistatic
results which provide a good reference for the full quantum calculations of the NM with
42
3.3
Plasmon modes of a nanomatryoshka
large gap, and thus no tunneling. The nonretarded approximation is well justified in
case provided small relevant dimensions of the structure. The classical results have also
the advantage of featuring well defined many-body plasmonic modes, not affected by
the interaction with single-particle electron-hole pair excitations which are typical for
the quantum TDDFT calculations in systems of small size like this [41, 125, 126]. In
Fig. 3.2(a) we show the absorption cross section per unit length of an individual cylindrical nanowire with external radius R = 61 Å (black dashed line). This would correspond
to the limiting case of a NM with zero width of the gap S = 0. Our calculations also show
the results for a cylindrical shell with internal radius R2 = 47.7 Å, and external radius
R3 = 61 Å, (blue line), and finally, the case of a cylindrical NM with the same shell as in
the previous case, and a core of radius R1 = 37 Å, i.e. a cylindrical (37, 47.7, 61) Å NM
(red line). In units of Bohr radius the dimensions of the system are (70,90,115) a0 2 . For
the individual nanowire, the TDDFT data (green line) is presented along with classical
results (dashed black line) allowing to access the quality of the classical Drude modeling
of the optical response in this case.
The absorption cross section of the individual cylindrical nanowire representative for
the core of the NM is dominated by the single plasmon resonance at surface plasmon
√
frequency ω− = ωp / 2, where ωp is the bulk plasma frequency in the Drude model of
dielectric function given by Eq. 1.2. As already pointed out, because of the nonlocal
screening and the small radius of the nanowire, in our TDDFT calculations ω− = 4.02 eV
is red shifted with respect to the corresponding surface plasmon frequency of the Na
metal, 4.16 eV [41, 123, 128, 131–135]. The distribution of the plasmon-induced charges
of this mode is schematically shown in Fig. 3.2(b). The optical absorption of the cylindrical shell is characterized by a low frequency bonding resonance ω − , and a high frequency
antibonding resonance ω + [125, 126] with plasmon charges separated, respectively, over
the entire shell or across each shell boundary as shown in Fig. 4.1(b). In the NM, the
ω− dipole mode of the core nanowire hybridizes with ω − and ω + dipole modes of the
−
+
nanoshell giving rise to the ω−
, ω−
, and ωc+ resonances. The lowest energy bonding
−
hybridized resonance ω−
at 1.1 eV is formed by the bonding shell mode ω − with an
+
admixture of the core ω− resonance. The most prominent ω−
absorption resonance of
the nanomatrioshka at 4 eV is formed primarily by the symmetric coupling of the ω −
core mode with the ω + shell resonance. Finally, the antisymmetric coupling of the ω−
core mode with the ω + shell resonance forms the anti-bonding resonance of the NM,
2
In practice, atomic units are the natural choice for quantum studies, and while in most of the cases
we give dimensions in Å for convenience of a broad audience, our TDDFT calculations use atomic units.
43
Chapter 3
ωc+ (here we use the terminology from Ref. [3] by Kulkarni et al.). As follows from
+
Fig. 3.2(b), the charge distribution of the ω−
mode corresponds to the parallel core and
shell dipoles so that this mode strongly interacts with incident electromagnetic wave.
−
The ω−
and ωc+ modes are characterized by the antiparallel core and shell dipoles, there-
fore the corresponding absorption resonances are weak in this case.
3.3.1.
Near field distribution
In the insets of Fig. 3.2(a) we present the near-field distribution of the plasmon resonances
of the NM. Owing to the opposite sign of the plasmon induced charges at the surface of
−
the core and inner surfaces of the shell, the lowest energy bonding hybridized mode ω−
is characterized by a strong field enhancement inside the core-shell gap. On the other
hand, because the core and shell dipoles are antiparallel, the fields are weak outside
−
the structure. Similar to ω−
, and because of the induced charge configuration, for the
+
ω−
resonance the fields in the gap are also strongly enhanced. For this mode the core
and shell dipoles are aligned. This leads to strong induced fields outside the structure
+
consistent with the most intense peak in the absorption cross section at ω−
. The core
+
character of the ω−
resonance clearly appears in the structure of the corresponding in-
duced fields showing a bright core region. The ωc+ mode is characterized by the opposite
sign of the core and shell dipoles and by the same sign of plasmon-induced charges across
the gap. The fields are thus low in the gap, core and vacuum regions. Strong plasmoninduced fields are only calculated inside the shell consistent with the shell character of
this plasmon mode. It should be noted that the character of the modes and their near
field distribution is analogous to that of a spherical NM [3, 114–119]. The resonance
energies are however different because of the different dimensionality of the problem.
√
Indeed, for a Drude metal, the quasistatic cylindrical core plasmon is at ω− = ωp / 2,
√
while the spherical core plasmon is at ω− = ωp / 3.
From a qualitative analysis of the induced charge distributions and near fields of the
modes, it is possible to presume that quantum effects might be relevant when the gap
separation between the core and the shell in the NM decreases. This situation can occur
by means of an increase of the core radius R1 . Analogous to the spherical NM case [3],
because of electron tunneling between the core and the shell, the gap becomes conductive
for small S (narrow gaps). In the touching limit (S → 0), a situation analogue to that
44
3.4
Quantum tunneling across the gap.
of the continuous metallic cylinder can be reached even prior to the direct geometrical
−
contact at R1 = R2 . In such a situation, the modes ω−
and ωc+ would disappear and
+
the mode ω−
would evolve into that of the metallic cylinder with a radius given by the
−
+
external radius of the shell R3 = 61 Å. Since ω−
and ω−
resonances are characterized by
a distribution of opposite charges across the gap, the effect of tunneling in these modes
can be expected to be stronger than for the ωc+ resonance. The above discussion is fully
confirmed by the quantum TDDFT results presented in the next section.
3.4.
In the left column of Fig. 3.3, we show the waterfall plots of the absorption cross section
of the NM per unit length. The results are shown as function of the frequency ω of
the incident radiation polarized perpendicular to the symmetry axis of the system. The
calculations were performed over a wide frequency range, and for various gap sizes, from
well separated core and shell, down to the situation of conductive contact at S = 0.
Different panels of the figure correspond to the results obtained within classical calculations using the Drude model of the metal permittivity (top), within the full quantum
TDDFT (center), and with the quantum corrected model (QCM) (bottom). The system
geometry is defined by (R1 , 47.7, 61) Å, where the core radius has been varied within
the limits 26.5 Å≤ R1 ≤ 47.7 Å, allowing to vary the size of the vacuum gap according
to S = 47.7 − R1 Å.
The gross features of the results are similar for the classical, quantum and QCM descriptions of the system. These results resemble those reported in the literature for similar
core-shell systems [3, 113–119]. With an increasing value of the core radius, R1 (decreasing S), the interaction between the induced charges across the gap increases, and the
−
pair of hybridized plasmons with shell character display red-shift (ω−
) and a blue-shift
(ωc+ ), respectively. Because of the mutual compensation between the core and the shell
dipoles for these modes [see Fig. 3.2(b)] their spectral features progressively loose their
−
intensity in the absorption spectrum. For R1 → R2 , the ω−
and ωc+ resonances can be
hardly distinguished in the spectrum. The differences between the quantum and the
−
classical calculations, particularly for the ω−
mode, cannot be observed in the left-hand
side spectra of Fig. 3.3, since their scale is set by the main peak in the absorption cross
+
section (the core resonance ω−
). For a vanishing gap, this prominent resonance evolves
into the dipolar resonance of the full metallic cylinder with radius R3 = 61 Å. It is
45
Chapter 3
R1(S)
S=0
S=40
R1(S)
S=0
S=40
R1(S)
S=0
S=40
Figure 3.3: Waterfall plot of the absorption cross section per unit length, σ of a cylindrical nanomatryoshka defined by (R1 , 47.7, 61) Å or equivalently (R1 /0.53, 90, 115) a0 ,
where the Bohr radius a0 = 0.53 Å calculated using classical electromagnetic theory
(top), TDDFT (center) and the Quantum Corrected Model (QCM) (bottom). Results
are given as a function of the frequency ω of the incident radiation for different core radii
R1 . The left panels show the results within a large ω range showing all the resonances
described in Fig. 3.2. R1 varies within the limits 50a0 ≤ R1 ≤ 90a0 (26.5Å ≤ R1 ≤
47.7Å) corresponding to gap separation distances, S, from 40 a0 down to 0, as indicated from bottom to up on the spectra. For clarity, a vertical shift is introduced to
each absorption spectrum. The curves are displayed in red every 10 a0 ≈ 5.3Å of
R1 -change. The lowest absorption spectrum in each panel corresponds to R1 = 50 a0
(S = 40 a0 ), and the red dashed curve on top (S = 0) is used as a reference for the
absorption spectrum of the solid metallic cylinder with external radius R = 115 a0
(61Å ). The plasmonic modes responsible for the peaks in the absorption cross-section
−
+
are labeled in each panel. ω−
stands for the bonding hybridized resonance, ω−
for the
+
hybridized resonance with a dominantly core character, and ωc for the antibonding
resonance of the nanomatryoshka. The right panels focus on results for core radii R1 in
the range of 78 a0 ≤ R1 ≤ 84 a0 (41.3Å ≤ R1 ≤ 44.5Å) at the frequency range of the
−
ω−
plasmon of 0.4 eV ≤ ω ≤ 1.4 eV . These are the conditions where the effect of the
−
resonant electron transfer on the bonding hybridized plasmon ω−
is most prominent.
The correspondence between the color used for the absorption spectra and the value of
the core radius R1 is given at the upper right panel of the figure. R1 is indicated in
units of a0 and the number in parenthesis gives the corresponding size of the gap, S,
between the core surface and the inner surface of the outer shell.
46
3.4
+
worth mentioning that for small R1 (large S), the TDDFT results for the core ω−
plas-
mon show a small (∼ 0.2 eV) red shift because of the electron spill-out effect and the
nonlocal screening [123, 128, 131–135]. Along with (slight redshift), for small R1 the
+
ω−
resonance features an additional broadening because of the increased Landau damp-
ing. It is also possible to notice some additional structures in the TDDFT absorption
cross-section due to the presence of electron-hole pair excitations [41, 125, 126]. The
left-lower panel of Fig. 3.2 shows that the QCM reproduces well the details of the spectra. For example, it captures the abrupt change of the main absorption peak at small
gap separation distances (spectra on the top), similarly to that present in the TDDFT
calculations, whereas the classical calculations show a smoother evolution. This abrupt
change at S ≈ 1 Å is linked with the lowering of the potential barrier separating the
core and the shell below the Fermi level. Under these conditions, electrons flow through
the gap quasi-freely forming a continuous metallic connection.
As we discussed above, the effect of electron tunneling is expected to be the strongest for
−
the ω−
resonance, since it is characterized by induced plasmon charges of opposite sign
across the gap. On one hand these opposite charges induce strong field in the junction
and thus (i) increase the coupling between the core and the shell plasmons; as well as
(ii) increase the tunneling current. On the other hand, this tunneling current, when
large enough, neutralises the charges across the gap. To reveal the role of quantum
tunneling in the NM gap, in the right panels of Fig. 3.3 we zoom into the spectral range
−
of frequencies of the ω−
resonance, showing the evolution of this lowest energy mode
as the gap becomes smaller. We compare results obtained with Classical Drude (top),
TDDFT (center), and QCM (bottom) approaches for values of core radius R1 in the
range of 78 a0 ≤ R1 ≤ 84 a0 (41.3 Å ≤ R1 ≤ 44.5 Å) corresponding to gap sizes in the
range of 6 a0 ≤ S ≤ 12 a0 (3.2 Å ≤ S ≤ 6.4 Å). As found in previous calculations of the
plasmonic dimer structures [1, 38, 39, 41, 42, 122], tunneling effects progressively appear
−
for narrow gaps. In the classical calculations, the ω−
resonance is always present albeit
being attenuated and red-shifted with increasing R1 (decreasing width of the gap S).
The quantum TDDFT results show a qualitatively distinct behavior. For the smallest
−
value of the core radius R1 = 78 a0 considered in Fig. 3.3, the ω−
absorption resonance
calculated with TDDFT agrees well with classical result. However, the resonance is fully
quenched for R1 = 82 a0 . For this very narrow gap of S = 8 a0 , electron tunneling across
the gap neutralizes the induced plasmonic charges of opposite sign at the facing surfaces
of the core and the shell. This leads to the disappearance of this bonding hybridized
plasmon. In contrast, in the classical calculations this gap mode is always present for
47
Chapter 3
any S 6= 0.
In the lower panels of Fig. 3.3, we can observe that the TDDFT results are well reproduced by the QCM calculations, including progressive broadening and final quenching
−
of the ω−
resonance for narrowing gaps due to the increasing resistive tunneling losses.
Since classical electrodynamic theory does not account for tunneling, it fails to reproduce
the full quenching of the gap plasmon prior to the direct geometrical contact between
the core and the shell. The failure of the classical theory and success of the QCM in
reproducing the TDDFT results can be clearly appreciated, for instance by comparing
the spectra in red line on the three right hand side panels of Fig. 3.3. Finally, the
high energy antibonding mode ωc+ is not so affected by quantum tunneling due to the
symmetric distribution of charges at both sides of the gap (see Fig. 3.2). Classical and
TDDFT calculations give practically the same results in this case for all separation distances (including the narrowest ones), therefore the QCM does not bring any essential
improvement for this mode.
3.4.1.
The near field enhancement
In Fig. 3.4 we show the field enhancement factor FEF for the field in the middle of the
gap, as obtained with the classical Drude, QCM and full quantum TDDFT approaches.
~
~ is the electric field measured inside the gap at
The FEF is defined as |E| , where E
|E~0 |
position R = (R1 + R2 ) /2 along the axis defined by the electric field vector of the incident plane wave E~0 . The FEF has been calculated for the incident electromagnetic
−
+
wave, resonant with the energy of the hybridized ω−
mode and with the ω−
. For a
size of the gap S, bellow 6 Å (core radius above 78 a0 ) the quantum TDDFT results
show distinct differences with respect to the classical predictions (area marked with a
shaded background in Fig. 3.4). The onset of the electron tunneling between the core
−
and the shell comes along with the disappearance of the ω−
resonance, and thus with a
pronounced decrease of the FEF. Eventually FEF for small gap sizes where the limit of
the homogeneous metallic cylinder is reached prior to direct physical contact between
the core and the shell at R1 = R2 . The QCM correctly reproduces the results of the
+
quantum TDDFT, including a decrease of the field enhancement at the ω−
resonance
for intermediate tunneling distances. Indeed, the resistive tunneling losses lead to a
48
3.4
Figure 3.4: Field Enhancement Factor FEF calculated in the middle of the gap
between the core and the shell (at R = (R1 + R2 ) /2) following the axis defined by
the polarization of the incident field. Dots represent the TDDFT results, dashed lines
represent results obtained using classical Drude calculations, and solid lines represent
the QCM results. Red color is used for the data at the frequency of the lowest energy
−
bonding hybridized resonance ω−
, and blue color for the data at the frequency of the
+
main absorption peak labeled as the ω−
resonance. The shaded background separates
the region where tunneling occurs from the classical region.
certain broadening of the plasmon peak which results in a moderate decrease of the corresponding field enhancement. For vanishing separation, S → 0, the junction becomes
conductive and FEF tends to the characteristic value of a homogeneous metallic cylinder
of radius R3 .
The ability of QCM to describe the quantum results has been reported in previous calculations for spherical and cylindrical dimer structures with gaps characterized by narrow
contact regions which allow for the presence of tunneling current. In these systems the
progressive disappearance of the bonding dipole plasmon mode and the appearance of
49
Chapter 3
the charge transfer plasmon mode prior to direct contact between the nanoparticles has
been addressed within the quantum TDDFT calculations. These features are correctly
reproduced with the QCM, while classical theories fail to address the spectral behavior, even qualitatively. Already at finite junction sizes the particles forming the dimer
are conductively connected showing the characteristic charge transfer plasmon modes
[78, 101–103, 105, 136]. In our calculations we observe the same physics, with the core
and the shell of the nanomatryoshka being conductively connected prior to the direct
contact between their surfaces, and thus effectively forming the uniform metallic cylinder. However, differently to plasmonic dimer structures, the tunneling current in the
case of a cylindrical NM flows over the whole extended gap region, a substantial dimensional difference that makes the NM an appropriate system to test quantum effects in
plasmonics.
3.4.2.
Plasmon modes of a spherical gold nanomatryoshka
The quantum effects described here for the cylindrical core-shell NM structure are similar
to those found by Kulkarni et al. in their TDDFT study of the spherical gold NM [3].
Therefore, we also analyze the performance of the QCM model in this spherical case.
We use the QCM to calculate the absorption cross section of a gold NM within the
non-retarded quasistatic approximation. This choice is justified considering the size of
the studied system (external radius below 8 nm), and it is consistent with non-retarded
calculation of the potentials employed in the TDDFT study. For the sake of comparison,
we use the Drude-like model for the dielectric permittivity of gold as developed in Ref.
[3] based on their quantum calculations of the optical absorption:
ωp2
,
ω(ω + iγ)
ε(ω) = ε∞ −
(3.4)
with ε∞ = 8, ωp = 9.07 eV, and γ = 0.27 eV. Within the QCM model, the effective
dielectric medium in the gap, εef f is defined by the frequency, and separation dependent
dielectric constant
εeff (S, ω) = ε∞ −
ωp2
.
ω(ω + iγeff (S))
where γeff (S) for gold is given by Eq. 3.3 with ∆ = 0.4 Å [1, 2].
50
(3.5)
3.4
0.06
σ (arb. units)
1
1.5 1.6 1.7 1.8 1.9 2.0 2.1
ω (eV)
ω+-
TDDFT
ω-0
0.1
σ (arb. units)
1
QCM
1.5 1.6 1.7 1.8 1.9 2.0 2.1
ω (eV)
0
1
1.5
2
2.5
ω (eV)
3
3.5
4
Figure 3.5: Absorption cross section, σabs , for a spherical gold nanomatryoshka as
a function of frequency ω. The results are normalized to the absorption maximum.
Different NMs are considered whose dimensions are defined by λ × (R1 = 8.5, R2 =
9.5, R3 = 15.9) Å where the values of λ ranges from 1 to 5, as displayed in the inset. R1
is the core radius. R2 , and R3 stand respectively for the internal and external radius
of the shell. Upper panel: quantum results from TDDFT calculations from Ref. [3] by
Kulkarni et al. Lower panel: current results based on the QCM. The bonding hybridized
−
+
resonance ω−
, and the main absorption peak of the core resonance ω−
are indicated in
the upper panel. The graphs in the insets show the respective zoom in of the low-energy
bonding hybridized resonance.
51
Chapter 3
Fig. 3.5 presents the frequency dependent optical absorption cross section, for five
spherical NM geometries (λ × (8.5, 9.5, 15.9) Å). These correspond to the scaling of
a basic NM structure given by (8.5,9.5,15.9) Å, with a scale parameter λ = 1, 2, 3, 4, 5.
This scaling [3] is implemented to rely on the scale invariance of the plasmon resonances
within the non-retarded limit of the classical electrodynamics. The absorption cross section should be identical for scalable systems with scale factor λ3 . Thus, any departure
from the universal behavior can be associated with a signature of a quantum effect. The
upper panel of Fig. 3.5 reproduces the quantum results of Kulkarni et al. and the lower
panel of Fig. 3.5 shows the absorption spectra obtained within our QCM calculation.
For large gap sizes (4 Å and 5 Å), tunneling across the gap is not efficient and the results
show the universal behavior pointed out above. When the gap size is reduced to 3 Å,
the electron tunneling through the gap increases enough to lead to a visible attenuation
−
of the ω−
resonance. Finally, similar to the cylindrical NM case, when the gap size is
reduced below 2 Å (4 a0 ), the tunneling current across the gap becomes large so that the
lowest energy bonding hybridized resonance is quenched. This change of the absorption
spectra exceeds the capabilities of the classical theory to treat such extreme gaps, as
pointed out in [3] by Kulkarni et al. However the TDDFT results are well reproduced
with the QCM, confirming the good performance of the latter in description of the optical properties of a spherical geometry with an extended tunneling contact.
3.5.
Summary
In conclusion, the work in this chapter, shows the applicability of the quantum corrected
model to the description of the optical properties and plasmonic modes in the systems
which show an extended tunneling contact region between metallic surfaces. This conclusion stems from the thorough comparison of the optical response calculated with the
QCM and with full quantum TDDFT for the case of a cylindrical nanomatryoshka,
where the metallic core and metallic shell are separated by a vacuum gap. We considered a metallic system made of Na because the free-electron character of the Na
valence electrons allows for a jellium model description. Reasonably large-size system
with well developed plasmonic modes can be addressed in this way at the fully quantum
level. Moreover, in this case the material permittivity can be well described with the
Drude model which eases the comparison between quantum, classical electromagnetic,
52
3.5
Summary
and QCM results.
We obtained that when the core-shell gap is reduced below 5 Å, the optical response is
determined by the quantum tunneling of conduction electrons across the potential barrier
separating the core and the shell. Our results agree with earlier calculations on plasmonic dimer structures. Specifically, when the width of the core-shell gap is reduced, the
lowest energy bonding hybridized plasmon mode disappears and the field enhancement
in the middle of the junction is quenched. The limit of the homogeneous metallic cylinder is reached prior to the direct contact between the core and the shell. The classical
local Drude description fails to reproduce the observed effects since it does not account
for tunneling. In contrast, the QCM results are found to be in good agreement with
the full quantum calculations. For large gap sizes, electron tunneling is negligible and
there is an overall agreement between TDDFT, classical Drude, and QCM calculations.
Therefore, as a whole, the QCM performs very well over the entire range of core-shell
gap sizes S studied here. We note however that the TDDFT results show some features
that are not reproduced with classical models: the size dependent frequency shifts of the
plasmonic modes due to the nonlocal screening, the size dependent broadening because
of the Landau damping and, finally, additional spectral features because of electron-hole
pair excitations.
We have also performed the QCM calculations of the optical response of the spherical
gold NM, where results of earlier quantum calculations were available [3]. The spectral
trends are similar to those found for the cylindrical sodium NM. The absorption spectra
obtained with QCM and with quantum calculations are in very good agreement. This
validates the applicability of the QCM to general systems characterized by extended
tunneling contact regions between metallic surfaces. Together with the earlier studies in
plasmonic dimers that showed narrow tunneling contacts, our results extend the range
of applicability of the QCM. Elucidating the main physics in tunneling plasmonic gaps
shows enormous potential with important consequences in the accurate description of
far- and near-fields in extreme morphologies, as well as in the control of non-linear effects
associated to ultranarrow gaps such as in rectification.
The results obtained in this chapter, prepare and validate theoretical tools to be used
in description of the optical properties of more complex realistic systems systems with
extended plasmonic gaps. These kind of structures comprise hundreds of thousands of
53
Chapter 3
electrons, so that the TDDFT calculations are out of reach. The QCM appears then
as a theory of choice allowing to calculate optical response with account for quantum
effects. Comparison of the experimental and theoretical data for the spherical NMs with
functionalized gaps is the subject of the next Chapter.
54
Chapter 4
Tunneling effects in spherical gold
nanomatryoshkas
As we have shown in the previous chapter, quantum effects like electron tunneling across
the small cavities in plasmonic systems play an important role in defining the optical
response of structures with subnanometer gaps. These effects can strongly alter both
their near- and far-field optical responses. In this chapter, we use the QCM to investigate plasmon coupling in spherical gold nanomatryoshkas with different core-shell
separations and the core-shell gap functionalised with different dielectric materials such
as SiO2 and self-organized molecular layer(s) SAMs. We compare theoretical results
with experimental data in order to elucidate the main physical mechanisms operating
in these systems. Here, we show that when the width of the gap decreases below 1 nm,
the near- and far-field properties can no longer be described by classical approaches but
require the inclusion of quantum mechanical effects such as electron transport through
the self-assembled monolayer junction. This is consistent with surface enhanced raman
scattering (SERS) measurements indicating strong electron-transport across the molecular junction. The model presented here, provides an estimate for the AC conductances
of molecules in the junction.1
1
This chapter, including graphs and text contents, corresponds mainly to the work published in
Nanoletters [5]. In this work, the experimental results were performed by L. Lin, M. Xiong, Z. Liu, S.
Wang, H. Xu, H. Gu, and J. Ye from School of Biomedical Engineering and Med-X Research Institute,
Shanghai Jiao Tong University.
55
Chapter 4
4.1.
Tunneling effects in spherical gold nanomatryoshkas
Introduction
As we have already discussed in this Thesis, one of the most interesting properties of
plasmon resonances is their associated near-field enhancement and confinement. The
cavity mode formed in a narrow gap, is determined by the coupling of neighboring
nanostructures and causes this confinement to be even more pronounced as well as more
sensitive to its dielectric environment [137–139]. This phenomenon has been usually
well explained by classical electromagnetic theory, which predicts strong electric field
enhancements around such structural features. However, as we have shown in Chapter
3 for subnanometer gaps, the quantum mechanical effects may invalidate the classical
description and lead to a reduction in the electric field enhancements [1, 54, 97, 140].
Different models have been then developed to account for electron tunneling between
nanoparticles and the corresponding modification of the nature of gap plasmonic modes
[38, 39, 65, 66, 141, 142].
Experimentally speaking, understanding of quantum plasmon effects has been hampered
by the difficulty of experimental realization of nanostructures with precise control of gap
widths in the subnanometer regime. To date, although various state-of-the-art techniques including high-resolution electron-beam lithography [3], electron-beam induced
manipulation [54], dual AFM tip approaching [1], and dropcasting [140, 142], have been
employed to fabricate nanoparticle dimer structures with subnanometer gaps, the fabrication of more complex system still remains challenging. A particularly promising
narrow gap system in this respect is the nanosphere on a metal surface which allows the
fabrication of large scale formation of a subnanometer gap reproducibly and uniformly
by using a monolayer of self-assembled molecules in between [5].
Here, we apply the QCM validated for the NM structures in Chapter 3 to explain
the experimental measurements of the optical response of the gold NMs composed
by spherical metallic cores surrounded by a concentric metallic shells separated by a
dielectric layer (see Fig. 4.1) [71, 143]. The core-shell gap can be tuned by controlling
the thickness of the dielectric spacer. For the narrow gaps below 0.4 nm the structures
are synthesized using SAMs as spacer layers. We show that the tunneling across the
spacer layer significantly modifies the absorption cross section and the local electric field
enhancements [144].
56
4.2
System and general considerations
Figure 4.1: (A) Geometrical illustration of a gold spherical nanomatryoshka (NM).
(B) Absorption cross-section calculated using classical electromagnetic theory for a
NM (r1 , r2 , r3 ) = (12.5, 13.5, 30) nm, with a SiO2 spacer layer. Insets represent the
schematic surface charge distributions for the modes at 532 nm (left) and 1057 nm
(right).
4.2.
System and general considerations
We present in Fig. 4.1.A the geometry of an Au spherical NM defined by the set of
structural parameters (r1 , r2 , r3 ) representing the radius of the inner Au core (r1 ) and
the inner (r2 ) and outer (r3 ) radii of the Au shell. The dielectric spacer between the
core and the shell consist either of a SiO2 layer or a monolayer of 1,4-benzenedithiol
(1,4-BDT) molecules. The larger gaps are realized by controlling the thickness of a
SiO2 layer inside the gaps, while the subnanometer gap is created using a self-assembled
monolayer (SAM) of 1,4-BDT molecules. The mechanism followed to synthesize the gold
NM’s is described widely in Ref. [6].
Previous results have indicated that classical electromagnetic theory is appropriate for
describing the optical response of NMs with gaps as small as 1 nm [71]. Fig.
4.1.B
displays the absorption spectrum of a NM of geometry (12.5, 13.5, 30) nm calculated
57
Chapter 4
using classical electromagnetic theory. The calculations were performed using a standard implemented Boundary Element Method (BEM) code to solve the electromagnetic
Maxwell equations2 . The empirical dielectric functions of Au and SiO2 were taken from
Johnson and Christy [30]. The refractive index of 1,4-BDT molecules was assumed as
1.4 [137], and the whole simulation region is defined with a background index of 1.33
corresponding to water. The calculated absorption spectrum shows a strong resonance
at 532 nm and a relatively weak resonance at 1057 nm, which can be explained within
the framework of plasmon hybridization [145]. The resonance at 1057 nm corresponds to
the excitation of the bonding hybridized plasmon of NM formed by the bonding dipolar
plasmon mode of the shell with admixture of the dipolar plasmon mode of the core. The
resonance at 532 nm can be assigned to a hybridized plasmon mode of the NM formed
from the dipolar mode of the core and the antibonding mode of the shell. The insets of
Figure 4.1.B illustrate the surface charge distributions of the two resonances. Computations based on the quantum mechanical modeling have predicted that the mode at 1057
nm is highly sensitive to quantum effects due to the high surface charge density and the
right charge symmetry in the nanogap [143, 144].
4.3.
Plasmon hybridization and quantum effects in NM of
different sizes
In order to investigate the plasmon hybridization and the importance of quantum effects in the NMs, we studied gold NMs with three size ranges of the core-shell gap: (i)
∼ 100 nm, (ii) ∼ 10 nm and (iii) sub-1 nm. All structures were synthetized by the group
of Jian Ye [6]. Fig. 4.2.A shows the schematic illustration of NMs with gaps in three size
ranges. The width of the gap is taken as the main parameter influencing the plasmon
coupling effects between the core and the shell. For example, a 100 nm gap size may
only induce a weak plasmon coupling because the decay length of near-fields normal to
the metal surface typically extends by only approx. 10 – 20 nm [146]. In contrast, a
∼ 10 nm gap size causes a strong coupling, hybridization, and near field enhancement.
When the gap size reaches the sub-nanometer scale, electron tunneling between the core
and the shell may occur and can influence both the near- and far-field optical properties,
2
Developed by Javier Aizpurua from the Theory of Nanophotonics Group at Materials Physics Center,
Donostia-San Sebastián, España and Javier Garcı́a de Abajo from Nanophotonics Theory Group at
Institut de Ciènces Fotòniques.
58
4.3
Plasmon hybridization and quantum effects in NM of different sizes
Figure 4.2: Optical properties of NMs of different geometries: (i) (25, 125, 150) nm
with a gap size of 100 nm and with SiO2 space layer; (ii) (40, 55, 68) nm with a gap
size of 15 nm and with SiO2 spacer layer; (iii) (10, 10.7, 25) nm with a gap size of
0.7 nm and with 1,4-BDT spacer layer. (A) Geometrical illustration, (B) TEM/SEM
images, (C) experimentally measured extinction spectra, and (D) calculated extinction
spectra using classical electromagnetic theory for the NMs (i-iii). Dashed lines in (C,D)
show the experimentally measured and calculated extinction spectra of aqueous Au
nanoparticles with a radius of 25 nm, respectively. (E) Simulated electric field intensity
(log |E/E0 |) distributions (top) and surface charge distributions (bottom) at resonance
corresponding to the modes 1-6, using classical electromagnetic theory. The SEM image
in (B) and the measured extinction spectrum in (C) for the NM (ii) were reproduced
from ref [143] for comparison between the NMs with different gap sizes.
as we discussed in the Chapter 3.
TEM image of the NM in Fig. 4.2.B shows the presence of SiO2 layer of thickness of
∼ 100 nm and of Au shell of thickness of 25 nm in average. The red-shift of 20 nm in
the extinction spectrum also verifies the presence of the coating of SiO2 layer on the Au
cores. The measured extinction spectrum of the synthesized NMs (i, Figure 4.2C) shows
two broad resonance peaks at 656 nm and 747 nm, respectively. The extinction spectrum
calculated with BEM (i, Fig. 4.2D) agrees well with experimental results, indicating that
the two peaks correspond to the hexapolar resonance of the outer Au shell for mode 1
and to a quadrupolar resonance for mode 2, as obtained from the calculated electric field
distributions and surface charge profiles (top in Fig. 4.2.E). The strong coupling of the
higher-order modes (mode 1 and 2) of the NMs to incident light is primarily attributed
to phase-retardation effects due to the relatively large NM size [115]. The weak electric
59
Chapter 4
Figure 4.3: (A) High-resolution TEM images and (B) histogram of different gap sizes
in NMs. All scale bars correspond to 5 nm. For panel B, the three gap-size peaks
fitted by Gaussian model correspond to i-iii in panel A and are centered at (i) 0.72 nm;
(ii) 1.24 nm; (iii) 1.76 nm. The insets in panel B illustrate the schematic hypothetical
configurations of 1,4-BDT molecules inside the gaps.
field enhancements in the gap region imply a rather weak plasmon coupling between the
core and the shell, and therefore the far-field optical spectrum of NMs (25, 125, 150)
nm is here dominated by the scattering spectrum of the outer Au shell. However, the
dipolar resonance mode (mode 1) of the Au cores is excited at 632 nm.
Experimental extinction spectrum of (40, 55, 68) nm nanomatryoshkas (gap size of
15 nm) from Ref. [119] (ii, Fig. 4.2.B) shows two intense and broad peaks at 622 nm and
943 nm (ii, 4.2C). The corresponding extinction spectrum calculated with BEM reproduces the experimental features well with two resonances at 592 nm and 940 nm, denoted
60
4.3
as mode 3 and 4 in Fig. 4.2.D. Mode 3 is the bonding mode resulting from the dipolar
core and bonding shell modes, and mode 4 is the bonding mode arising from the core
and the antibonding shell dipolar modes, as confirmed by the calculated surface charge
distributions (Fig. 4.2.E). For mode 4, the pronounced electric field enhancements also
suggest a significant coupling between the core and the shell plasmons optical response
of NMs [147]. In addition, we note that the line width of the experimentally measured
plasmon resonances is broader than those of the calculated ones, which is most likely due
to the inhomogeneous effect from the polydispersed particle size of the synthesized NM’s.
In order to obtain NMs with subnanometer gaps, a self-assembled monolayer of 1,4-BDT
molecules was used as a spacer to uniformly and reproducibly control the core-shell separation [5]. A TEM image of NMs with a geometry of (10, 10.7, 25) nm is shown in
Fig. 4.2.B (iii). The core-shell separation was calculated as 0.72 nm on average, based
on the high-resolution TEM images of more than 60 nanoparticles (Fig. 4.3). Such NMs
were synthesized in a layer-by-layer process, starting with the preparation of 20 nm diameter Au cores [30] (see [5] for details). Assuming that 1,4-BDT has a perpendicular
orientation relative to the Au surfaces inside the gap, the gap width is given be the
thickness of 1,4-BDT monolayer. Using the known bond lengths, bond angles, and the
approximate distance between the sulfur atom and the h111i gold surface as calculated
with density functional theory [148–150], the gap width can be estimated to be in the
range of 0.8 nm to 1 nm for monolayer coverage. However, different gap sizes were
also observed in high-resolution TEM images (Fig. 4.3A) and a series of gap sizes were
carefully measured to make a histogram (Fig. 4.3B). The first peak is centered at a
gap size of 0.72 nm, which likely corresponds to a slightly inclined 1,4-BDT configuration with an angle of approximately 25◦ -45◦ with the surface normal for low surface
coverages. Such a tilt is in agreement previous estimates and calculations [140, 150, 151].
The second and third peaks in the histogram are centered at gap separations of 1.24 nm
and 1.76 nm, which most likely corresponds to a dilayer and a trilayer of 1,4-BDT
molecules, respectively (insets in 4.3.B) [150]. In addition to previous estimates by ellipsometry and calculations, TEM imaging of various gap sizes in NMs offers a direct
observation of different thicknesses of 1,4-BDT molecular films on a Au surface. It has
been noted that a 0.62 nm gap in NMs was reported previously and the authors had
attributed the shrinkage of the 1,4-BDT monolayer to the compression of the molecules
between the core and shell, proved by the up-shifts of the phenyl ring stretching mode
61
Chapter 4
Figure 4.4: (A) Extinction spectra, (B) electric field distributions, and (C) electric
field intensities at point a and b for a NM(10, 10.7, 25) nm calculated by QCM using
conductive molecule junction (0.08 G0 per molecule) and classical theory. Electric
fields were calculated at the wavelengths of λ1 = 511 nm, λ2 = 532 nm, λ3 = 785 nm,
λ4 = 895 nm, λ5 = 1100 nm. Inset in panel C indicates the position of point a (0.1 nm
apart from the outer surface of Au shell) and b (0.1 nm apart from the surface of Au
core). (D) SERS spectra of NMs (10, 10.7, 25) nm with 1,4-BDT spacer layer excited
by 532 nm and 785 nm laser.
and the CH bending mode in Raman spectra [143]. However, this blueshift of Raman
bands was not observed in the measurement showed.
Due to their relatively small size, the extinction spectrum of such NMs is primarily
dominated by the absorption cross-section as we show in Fig. 4.5. In this figure, the
62
4.3
Figure 4.5: Absorption, scattering, and extinction cross section of a NM with a
dimension of (12.5, 13.5, 30)nm.
extinction spectrum of a (10, 10.7, 25) nm NM is obtained using classical electromagnetic
theory (numerical BEM). The results show a noticeable resonance at 526 nm (mode 5)
and a weak resonance at 1090 nm (mode 6) in Figure 4.2D, which seems to behave
similarly to the plasmon hybridizations of mode 3 and 4, due to identical surface charge
patterns and near-field enhancements (bottom in Fig. 4.2E). In contrast to the larger
(40, 55, 68) nm NM, the narrower gap leads to stronger plasmon hybridization and energy
level splitting, inducing a spectral red-shift for the low energy mode and a blue-shift to
the high energy mode. However, the experimental extinction spectrum (iii, Fig. 4.2.C)
shows only one pronounced resonance at 539 nm, in direct contrast with the prediction
by classical theory in Fig. 4.2.D (iii). Quantum mechanical modeling has predicted the
disappearance of the lower energy mode (mode 6) as a direct manifestation of quantum
tunneling gap below 0.4 nm [144]. However, the core shell-gap of the present system
is too wide for quantum tunneling to occur between two gold surfaces. In fact, in the
present system the tunneling is due to the self-assembled molecular monolayer in the
plasmonic gap, where the 1,4-BDT molecules play the role of molecular wires bridging
Au electrodes as has been extensively studied theoretically and experimentally in the
63
Chapter 4
context of molecular electronics [151–161]. Recent studies have demonstrated that, indeed, the optical response of plasmonic systems can be affected by electron tunneling
through molecular monolayers bridging narrow gaps between the particles [140, 162].
Similar to the metal-to-metal tunnel junction, the tunneling current through the molecular tunnel junction short circuits the junction, neutralizes the plasmon induced charges
across the gap, and leads to the disappearance of the corresponding plasmonic mode
from the optical spectra. To support this idea, we show the measured and calculated
extinction spectra for homogeneous Au NP’s with the same outer diameter as the NM
(50 nm) (dashed lines in Fig. 4.2C, D). The measured spectrum of these solid particles
agree almost perfectly with that of 1,4-BDT embedded NM’s as if the gap would be
absent, which points at the presence of tunneling in the (10, 10.7, 25) nm NMs.
4.3.1.
Quantum tunneling in NM and subnanometric molecular layers
To model the effect of quantum tunneling on the optical properties of the NM, we
modified the QCM presented in the previous chapter in order to account for electron
transport through the molecular layer. We model the 1,4-BDT SAM using a permittivity
[162] given by:
εSAM = ε∞ + i
4πσ
,
ω
(4.1)
where ε∞ = 2. Assuming a linear optical potential variation across the gap, the conductivity σ of the molecular layer is given by
σ=
αG0 d
Σ
(4.2)
Here G0 = 7.748 × 10−5 S is the quantum of conductance, d is the width of the gap,
and Σ = 0.22 nm2 is the typical surface area per molecule in a dense SAM [163]. The
coefficient α gives the conductance of the single molecule G in units of G0 : G = αG0 .
Fig. 4.4.A shows the extinction spectra of a (10, 10.7, 25) nm NM calculated using both
QCM and the Boundary Element Method (BEM) [164]. In QCM calculations we use
α = 0.08, which is the lower limit of the conductance per molecule at which the plasmonic mode at ∼ 1100 nm disappears from the optical spectrum (see also Fig. 4.4).
The results show excellent agreement between the QCM and measured spectrum with
a strong dipolar mode around 520 nm but no sign of the longer wavelength hybridized
mode which is present around 1100 nm in the classical approach.
64
4.3
Figure 4.6: (A) Extinction spectra, (B) electric field distributions, and (C) electric
field intensities at point a and b for a (10, 10.7, 25) nm nanomatryushka calculated by
QCM using conductive molecule junction with different conductances (G0 ). Inset in
panel A is a zoom-in around high wavelength region. Electric fields were calculated
at the wavelengths of 785 nm and 1100 nm. Inset in panel C indicates the position of
point a (0.1 nm apart from the outer surface of Au shell) and b (0.1 nm apart from the
surface of Au core).
65
Chapter 4
Several remarks are in order with respect to these results: (i) As we will also discuss
in connection with Fig. 4.6, G = 0.08G0 , provides a lower estimate for the conductance
of the 1,4-BDT SAM needed to quench the plasmon resonance at ∼ 1100 nm. Since
σ ∼ α/Σ, less dense molecular layer than supposed here will require higher conductance
per molecule to reach the same effect. (ii) The present value of the molecular conductance is within the range of the experimental and theoretical estimates for electron
transport in individual 1,4-BDT molecules [151–161], particularly taking into account
that the inclined configuration of SAM tends to increase the conductance values [158–
161]. (iii) For the laser assisted transport in molecular junctions the ac conductance can
be significantly higher that the dc conductance measured in break junctions, in particular when photon absorption opens new conductance channels through molecular orbitals
[96, 165, 166].
Fig. 4.4.B shows the electric field distributions calculated by QCM and the classical
approach for wavelengths of 511, 532, 785, 895 and 1100 nm. The classical approach
predicts a large field enhancement in the interior gap for the 1100 nm mode while no such
field enhancement is present in the QCM where the tunneling current short-circuits the
gap. The distribution of electric field maps indicates that within the framework of classical theory and QCM, the near-field enhancement on the outer shell surface reaches the
maximum at the hybridized dipolar resonance at short wavelength (532 nm). This qualitative information from near field maps is further analyzed in Fig. 4.4.C where we plot
the electric field enhancement at the local points a and b (see the inset in Fig. 4.4.C). For
example, at point a, both within the QCM or classical theory, the electric field enhancement maintains a value below 5 for all wavelengths with a maximum of 4.7 at 532 nm,
which is very similar to the response of a solid spherical Au nanoparticle. For point
b, classical theory predicts that the field enhancement increases when the wavelength
becomes longer with a value below 5 at 511, 532, 785, and 895 nm and a maximal value
of ∼13 at 1100 nm; while QCM shows constant field enhancements for all wavelengths
with a maximal value below 3. Note the remarkably different near field behavior in
plasmonic gap (for point b) at longer wavelength, where strong near field enhancement
calculated with classical theory is not supported by QCM.
In Fig. 4.6.A, we show the extinction spectra the NM (10, 10.7, 25) nm calculated by
QCM as a function of junction conductance per 1,4-BDT molecule. As the conductance
per molecule of the junction increases, the resonance at the longer wavelength gradually
66
4.4
SERS measurement
decreases and finally vanishes for the largest conductance of 0.08G0 , while the resonance
at the visible range remains constant at around 520 nm. Thus, we estimate 0.08G0 as
a lower limit for conductance per 1,4-BDT molecule needed to explain the measured
spectra. Fig. 4.6.B and 4.6.C show the corresponding NM’s electric field maps and the
electric fields at the local points at the junction (point a) and outer shell surface (point
b) at the wavelengths of 785 and 1100 nm with different junction conductances (see inset
in Fig. 4.6.C). The results clearly indicate that the variation of junction conductance
has much more dramatic impact on the near-field enhancement in the junction for the
1100 nm mode compared to the 785 nm mode: electric field enhancements decreases from
∼7 to ∼2.5 when the conductance increased from 0.02 to 0.08G0 . In contrast, the variation of junction conductance has almost no influence on the near-field enhancement at
the outer shell surface. With QCM modeling, the strong dependence between the junction conductance and (near and far- field) optical properties of the NM (10, 10.7, 25) nm
can be explained by the shortcircuiting of the gap. The modeling developed here also
offers an approach to quantitatively estimate molecular conductance based on the farfield optical spectral measurement.
4.4.
SERS measurement
Surface-Enhancement Raman Scattering (SERS) is a powerful probe of local field enhancements and charge transfer (CT) effects [55, 167]. In addition to the numerical
calculations developed up to this point, we present SERS measurement on NM’s, that
were performed by the group of Jian Ye and presented jointly in [6], in order to complete
the study of the near-field enhancement and electron transfer in the molecular junctions.
Fig. 4.4.D shows the SERS spectra of (10, 10.7, 25) nm NM’s on a silicon substrate, where
strong characteristic Raman bands of 1,4-BDT molecules at 729, 1055, 1551 cm−1 are
observed when excited by 785 nm laser, and no significant bands are observed when
excited by a 532 nm laser (the strong band at 900 – 1000 cm−1 in both spectra comes
from the exterior silicon substrate). The dramatic difference between the Raman intensities of the interior 1,4-BDT molecules excited by 785 nm and 532 nm lasers can
not be explained by the near-field enhancement effect since the QCM-calculated electric
field intensities in the junction at 532 nm and 785 nm are almost identical and very
weak (Fig. 4.4.C). We can assert that the electron transport associated with the charge
transfer is the main enhancement mechanism that determines the SERS signal in the
67
Chapter 4
Figure 4.7: (A) TEM images, (B) experimentally measured and (C, D) calculated
extinction spectra using (C) QCM and (D) classical electromagnetic theory for NMs
(i-iv) with 1,4-BDT spacer layer and with different dimensions: (i) (12.5, 13.2, 31) nm;
(ii) (14, 14.7, 37) nm; (iii) (18, 18.7, 41.5) nm; (iv) (21, 21.7, 50) nm.
(10, 10.7, 25) nm NMs system, taking into account measurements and calculations made
in 4-methylbenezenethiol (4-MBT) molecules, which have a very similar molecular structure to 1,4-BDT molecules [5] (not shown here).
We complete our discussion of the quenching of gap plasmons in 1,4-BDT embedded
NM’s, with a study of additional NM’s of different geometries at subnanometric gaps
(Fig. 4.7.A): (i) (12.5, 13.2, 31) nm; (ii) (14, 14.7, 37) nm; (iii) (18, 18.7, 41.5) nm; (iv)
(21, 21.7, 50) nm. In each of these structures, the core-shell separation is 0.72 nm corresponding to the molecular length of 1,4-BDT. All measured extinction spectra of NMs
(i – iv) display only one resonance peak at the visible regime (Fig. 4.7.B). The resonance peak red-shifts from 544 nm to 585 nm as the size of the particle is increased
from 62 nm to 100 nm thus behaving as a homogeneous spherical Au NP. The QCM
extinction spectra (Fig. 4.7.C) are in excellent agreement with the measured spectra. In
contrast classical theory (Fig. 4.7.D) predicts additional longer wavelength modes. This
agreement clearly supports our conclusion of strong electron tunneling effects through
the subnanometric molecular junction of NMs.
68
4.5
4.5.
Summary
Summary
In conclusion, using classical electromagnetic theory and QCM model approach we have
theoretically investigated the optical response of Au nanomatryoshkas with core-shell
gaps ranging from 100 nm to the subnanometer regime and compared theoretical results
with experimental obtained on the systems. Our results confirm importance of the
tunneling phenomena in subnanometric gaps. As the width of the gap is decreased, the
hybridisation between core and shell plasmons increases. For subnanometer size gaps,
the electron tunneling across molecular junctions between the core and the shell becomes
important. It results in the charge transfer screening of the hybridized modes. Our
theoretical QCM simulations successfully reproduce the measured far-field spectra and
are further utilized for predicting the near-field enhancement in the molecular junction.
Electron tunneling in molecular junctions also manifests itself in SERS measurement
of the 1,4-BDT molecules present in the junction. The quantum properties exploited
in this joint work can provide a totally new insight on the ongoing development of the
strategies for optimal plasmonic SERS substrates with narrow junction gaps. The link
between plasmon modes and tunneling trough SAMs in the plasmonic gap revealed here
can provide a way to access the molecular conductance in the nanojunctions.
69
Chapter 5
The ability of localized surface plasmons to squeeze light and engineer nanoscale electromagnetic fields through electron-photon coupling at dimensions below the wavelength,
has turned plasmonics into a driving tool in a variety of technological applications, targeting novel and more efficient optoelectronic processes. In this context, the development
of active control of plasmon excitations is a major fundamental and practical challenge.
In this chapter, we propose a mechanism for fast and active control of the optical response
of metallic nanostructures, based on exploiting quantum effects in subnanometric plasmonic cavities. By applying an external DC bias across a narrow gap (in a core-shell
structure or in a plasmon dimer), a substantial change of the tunneling conductance
across the junction can be induced at optical frequencies, modifying in a reversible
manner the plasmonic resonances of the system. We demonstrate the feasibility of the
concept with proof-of-principle calculations performed using the time dependent density
functional theory (TDDFT). The results presented here, show that metal nanoparticle
plasmonics can benefit from the reversibility, fast response time, and versatility of an
active control strategy based on applied bias, establishing a platform for many practical
applications in optoelectronics1 .
1
This chapter, including graphs and contents, corresponds mainly to the work published in Science
Advances [6]. In this team, D. C. Marinica has performed the TDDFT calculations for dimers and
Mario Zapata for all the NM configurations, under supervision of Andrei Borisov. P. Nordlander, A. K.
Kazansky, P. M. Echenique and J. Aizpurua contributed to succesfull discussion.
71
Chapter 5
5.1.
Introduction
Advances in nanofabrication techniques[168, 169] allow to engineer plasmonic response of
the metal nanoparticles and nanoparticle assemblies and fully exploit strongly enhanced
near fields and scattering resonances in far fields in such systems[15, 16]. Sensors[17,
168], nanoantennas allowing single molecule detection[18], medical research[20], information transfer[21], single photon generation[22], exaltation of non-linear effects[23],
photochemistry[24], heat generation and hot electron injection[25], plasmon rulers[88]
constitute an incomplete list of applied and fundamental research lines linked to light
manipulation at subwavelength scales using plasmonic excitations. In this context the
importance of an active control over plasmonic properties of a system can not be underestimated. While at present the plasmon response is mainly tuned during the fabrication procedure via shape and material engineering, or alternatively, by the choice of
a particular dielectric environment[170, 171], it is highly desirable to have fast, progressive, and reversible procedures at hand to control this response. Recent experimental
developments trace several possibilities for such active control that can be achieved, for
instance, using flexible substrates [45, 172]; liquid crystal environment [46, 47]; tunable
molecular layers [173, 174]; electrically induced thermal heating [49, 175]; all optical
modulation using quantum dot arrays [50], or excitation of the free carriers [51, 52].
The latter allows to access typical electronic time scales of the order of fs (10−15 s), however this technique requires an incident laser of sufficient power to produce appreciable
effects. In this context, electrical control of the plasmonic response via applied bias, as
implemented at infrared and terahertz frequencies by graphene structures [53, 176, 177],
appears extremely attractive[178] offering the possibility of control at short timescales,
flexibility, and simplicity. However, practical realisations are limited to-date in to the
THz or mid-IR regime because of the low doping concentrations that can be achieved
electrically. The latter is also a bottleneck for applying the same strategy to the 3D bulk
plasmonic metal nanostructures with high free-electron density and plasmon response at
optical frequencies. So far, only solutions based on electrochemistry have been proposed
in this case [179–181].
A number of recent experimental and theoretical studies on nanoparticle dimers with sub
nm gaps [1, 38, 39, 41–43, 54, 55, 95–97, 122, 159, 182, 183] have demonstrated that,
when efficient, quantum tunneling through the potential barrier separating nanoparticles, strongly modifies the optical response of the entire system. Under tunneling
72
5.2
Model and qualitative aspects
conditions, the near field enhancement in the gap is reduced, absorption resonances owing to plasmonic modes of the capacitively coupled dimer are progressively attenuated,
and charge transfer plasmon modes of the conductively coupled dimer emerge prior to
the direct geometrical contact between the surfaces, as we widely discused in the last
two chapters. In the tunneling regime, the conductivity (or, equivalently, conductance)
across the junction is a key parameter defining the optical properties of the nanostructure. So far, the tunneling through the junction, and thus the plasmonic response, has
been modified by changing the width of the gap[54, 95, 96, 182] or by using plasmonic
gaps bridged by self assembled molecular layers[97, 159]. We develop a completely new
strategy and show that the bias voltage dependence of the electron tunneling through
narrow gaps can be used for active electrical control of the plasmonic response of the embedding metallic nanostructures. Indeed, similar to scanning tunneling microscopy[184–
186] at DC, the tunneling barrier and thus the conductance through the junction can
be controlled not only by the variation of the junction size, but also by an applied bias.
5.2.
Fig. 5.1 sketches the concept of the proposed control strategy. As shown in Fig. 5.1(a),
in the geometrical tuning of the plasmon response, change of the width of the gap
modifies the tunneling barrier and thus the conductance of the junction. For the fixed
geometry, as shown in Fig. 5.1(b), the modification the tunneling barrier can be produced
applying bias between metal nanoparticles separated by narrow gaps. As a consequence,
the conductance across the junction at optical frequencies, and therefore the plasmonic
modes of the nanostructure, can be modified in a controllable way. This allows to achieve
an active and flexible control of the optical response of the system at the fast (typically
ps 10−12 s) time scales needed to bias the system.
In order to demonstrate the sought controllable modification of the optical response, we
perform a proof-of-principle calculation for two representative systems characterised by
the presence of a narrow plasmonic gap: (i) the cylindrical core-shell nanomatryoshka
(NM), and (ii) the spherical plasmonic dimer, as schematically shown in Fig 5.2(e) and
(f), respectively. The NM (R1 , R2 , R3 ) consists of an infinite cylindrical metallic core
of radius R1 , and a coaxial cylindrical metallic shell characterized by an internal radius
R2 and external radius R3 , using the standard notations introduced in Chapter.3. The
spherical plasmonic dimer is formed by two identical spherical metallic particles of radius
73
Chapter 5
a)
b)
vacuum level
w
EF
vacuum level
w
Jw
EF
EF
Jdc
Jw
U
EF
S2< S1
U
S1
S
Figure 5.1: Schematic representation of the concept underlying the active control
strategy. The evolution of the tunneling barrier separating two metal surfaces across
the junction is shown for two situations. a) The junction width is reduced from S1
to S2 . The Fermi levels EF of the left and right leads are aligned (shown with blue
lines). b) A bias U is applied to the left lead, while the width of the junction S is kept
fixed. The shaded gray (red) area represents the tunneling barrier for the electrons at
the Fermi energy of the right lead before (after) modification of the tunneling barrier
by the reduction of the width of the junction in panel a), or by the applied bias in
panel b). The presence of the incident electromagnetic field at frequency ω induces a
modulation of the one-elecron potentials, as represented with the vertical arrow on the
top left corner of the panels. Horizontal green and blue arrows show, respectively, the
AC current Jω due to the optical potential, and the dc current JDC due to applied bias
U.
R separated by a narrow gap of width S. Both systems exhibit distinctive plasmonic
modes characterized by the optical fields strongly enhanced in the gap. These modes
are thus particularly sensitive to the tunneling through the junction [3, 38, 39]. Indeed,
when the tunneling current is established it partially neutralizes the induced charges
of opposite sign at facing surfaces across the junction, thus altering the near fields and
electromagnetic couplings between the individual parts of the composite structure.
We identify the plasmon resonances sensitive to the AC current across the gap for
the cases of the nanomatryoshka and the dimer in their absorbance spectra showed in
Fig. 5.2(a) and Fig. 5.2(b). We use classical electromagnetic calculations in the quasistatic approximation, in order to obtain these spectra. This approach is well justified
because the relevant system sizes are much smaller that the wavelength of the incident
electromagnetic wave, and because the tunneling is negligible for the geometries with
8.5 Å wide gaps as considered in Fig. 5.2. Without loss of generality a vacuum gap is
74
5.2
Figure 5.2: Systems studied. Panels a), c): optical response of a cylindrical core-shell
NM characterized by the radius of the core R1 , and internal R2 and external R3 radii of
the shell. Results are shown for the (R1 = 39.2 Å, R2 = 47.7 Å, R3 = 61 Å) NM with
the core shell gap S = R2 − R1 = 8.5 Å. Panels b), d) show the same information for
a plasmonic dimer formed by the two spherical nanoparticles with radius R = 21.7 Å
separated by the gap of the width S = 8.5 Å. Panel a) shows the absorption cross section
per unit length σ/l for cylindrical NM. The absorption resonances are labelled according
−
+
to the underlying plasmonic modes. ω−
: bonding hybridized plasmon, ω−
: resonance
+
with core character, and ωc : nonbonding mode with shell character. Panel c) shows
the near field distribution for an incident plane wave polarised as sketched with blue
−
arrow and resonant with the ω−
mode of the NM. Panel b) shows the absorption cross
section σ for the spherical dimer. The absorption resonances are labelled according
to the underlying plasmonic modes. ωd : bonding dipolar plasmon, ωq the bonding
quadrupolar plasmon. Panel d) shows the near field distribution calculated for an
incident plane wave polarised as sketched with the blue arrow and at resonance with
the ωd plasmon mode. In the panels a) and b) the resonances most sensitive to tunneling
and thus subject to active control are marked with red arrows. Panels e) and f) sketch
the cylindrical and spherical dimer systems under bias, respectively.
75
Chapter 5
assumed in all calculations, nevertheless the possibilities offered by the dielectric filling of the gap are discussed below in this Chapter. In the case of the cylindrical NM
[Fig. 5.2(a)], the gap plasmon most sensitive to the tunneling is a bonding hybridized
−
mode ω−
associated with the low energy absorption resonance as already discussed
in Chapter. 3. For the spherical dimer [Fig. 5.2(b)], the gap plasmon most sensitive to the tunneling is a bonding dipole plasmon ωd widely discussed in the literature
[1, 38, 39, 41, 183]. This mode is associated with the main absorption peak. The in−
tensities of the resonant near fields for the ω−
and ωd gap plasmon modes of the NM
and spherical dimer are shown in Fig. 5.2(c) and Fig. 5.2(d). As pointed out, these
absorption resonances marked with red arrow in Fig. 5.2 are very sensitivity to electron
tunneling and thus offer a great opportunity to exploit our concept of quantum control
with an applied bias.
5.3.
TDDFT framework
We use the Kohn-Sham (KS) scheme within Time Dependent Density Functional Theory
(TDDFT) to perform the quantum-mechanical study of the electron dynamics triggered
in plasmonic nanoobjects by an external perturbation. Along with calculation of the
linear response TDDFT allows to follow the real time evolution of the electron densities
currents and fields in the system. The TDDFT is a well established state-of-the-art
ab initio approach for the treatment of both linear and nonlinear response to optical
excitations in molecules, clusters, and solids[124, 187, 188]. The cylindrical core-shell
nanomatryushka (NM), as well as spherical dimer (SD) are treated within the jellium
model (JM) detailed for these geometries in earlier publications [39, 42, 189]. The ionic
3 −1
cores are represented by uniform background density of positive charge n+ = 4π
3 rs
(we use atomic units unless otherwise stated). The Wigner-Seitz radius rs is set equal to
4 a0 (Bohr radius a0 =0.53 Å) corresponding to the Na metal which is a prototype system
for the jellium description. In the case of the (R1 , R2 , R3 ) cylindrical nanomatryoshka
(NM) we use variable R1 , and fixed R2 = 90 a0 (' 47.7 nm), and R3 = 115 a0 (' 6.1
nm). Here, to define the structure we use the same notations as in Ref. [3] and descibed
in Chapter. 3. Thus, depending on the geometry, the system comprises up to 620 conduction electrons per unit length. In the case of the spherical dimer each nanoparticle
has a closed shell structure and consists of Ne = 1074 electrons. The cluster radius is
1/3
R = Ne
rs = 40.96 a0 (' 21.7 Å). In all cases considered here the Fermi energy of the
76
5.3
TDDFT framework
ground state system is EF ≈ −2.9 eV with respect to the vacuum level.
Within the TDDFT the time-dependent electron density is given by
n(r, t) =
X
|ψj (r, t)|2
(5.1)
j∈occ
with the summation running over all occupied KS orbitals ψj (r, t). The time evolution
of KS orbitals is given by the time-dependent Scrödinger equation:
i
∂ψj (r, t)
= {T + Veff (r, t; n(r, t))} ψj (r, t).
∂t
(5.2)
Here, T = − 21 ∆ is the kinetic energy operator, and Veff (r, t; n(r, t)) is the effective KS
potential that depends on the electron density. It is thus generally time-dependent
through the time-dependence of n(r, t). Veff (r, t; n(r, t)) comprises three terms:
Veff (r, t; n(r, t) = VH (r, t; n(r, t)) + Vxc (r, t; n(r, t)) + Vext (r, t).
(5.3)
The first term in Eq. (5.3) is the Hartree potential for the electron interaction with
total charge density n+ − n(r, t). We use the non-retarded approximation which should
be valid considering the electric field polarisation and small radii of the NM and SD
considered here. The second term in Eq. (5.3) is the exchange correlation potential.
We use the adiabatic local density approximation (ALDA) described in [124, 188, 190]
with exchange correlation kernel of O. Gunnarsson and B. I. Lundqvist [191]. Finally,
the last term describes external perturbation which can be the electromagnetic field
and/or applied bias. With KS orbitals represented on a mesh in cylindrical coordinates,
Eqs. (5.2) are solved using real-time propagation similar to that used in the one-electron
wave packet propagation in cylindrical coordinates as detailed in [192, 193].
The linear properties are obtained from the response to a weak impulsive potential. To
this end we set: Vext (r, t) = γ p̂rδ(t), where γ is a small constant. The unit length vector
p̂ defines the polarization direction of the incident linearly polarized plane wave, so that
p̂r is the coordinate along the polarization direction. For the cylindrical core-shell NM p̂
is perpendicular to the symmetry axis, and it is along the symmetry axis for the spherical
dimer. The Fourier time-to-frequency transform of the near fields and of the induced
dipole moment allows us to calculate the frequency-resolved fields in the plasmonic gap
as well as the optical absorption cross-section given by σ(ω) =
77
4πω
c Im {α(ω)}.
Here
Chapter 5
α(ω) is the dipolar polarizability of the system. To converge the results, an artificial
frequency broadening of 0.15 eV is used, as typical for the TDDFT calculations on small
systems [38, 39, 122, 194]. For the small system sizes considered here the absorption
and extinction cross-section can be considered to be equal.
The initial conditions for the time dependent Eqs. (5.2), ψj0 (r) ≡ ψj (r, t = 0), correspond
to the K-S orbitals of the ground-state system, which implies that prior to using TDDFT
one performs the standard density functional theory (DFT) study. The K-S orbitals
ψj0 (r) satisfy the stationary Kohn Sham equations:
0
T + Veff
(r; n) ψj0 (r)
=
n(r)
=
Ej ψj0 (r),
X ψj0 (r)2
(5.4)
j∈occ
where Ej is the energy of the K-S orbital. The effective potential
0
0
(r; n) = VH0 (r; n) + Vxc
(r; n)
Veff
is time independent. We use superscript
0
(5.5)
to refer to the ground state system.
While the ground state electronic properties of the spherical dimer have been shown in
number of publications [38, 39, 122, 194], the NM is much less discussed [3]. In Fig. 5.3
we show the ground state electron densities and effective potentials Veff calculated with
DFT for the cylindrical NM with geometry given by (R1 , 90 a0 , 115 a0 ) or, equivalently,
(R1 , R2 = 47.7 Å, R3 = 61 Å). Results are presented for two values of the core radii
R1 = 60 a0 (31.8 Å), and R1 = 82 a0 (43.5 Å) corresponding to the gap size S = R2 − R1
equal to S = 30 a0 (15.9 Å) and S = 8 a0 (4.2 Å) respectively. In the first case the
potential barrier between the core and the shell is wide and the core and the shell are
coupled only capacitively with negligible tunneling through the gap. In the second case,
the tunneling contact through the narrow potential barrier is established. Along with
evolution of the potential barrier, one can also notice an extension of the electron density
profile beyond the jellium edge. This is particularly well seen for the external surface
of the NM, where the electron density extends outside the boundary located at R3 =
115 a0 (61 Å). The spill out of the electron density outside the nanoobject boundaries
results in a red shift of the dipolar plasmon frequency as compared to the prediction
of the classical theory [56, 122, 195]. Thus, for the individual spherical nanocluster
78
5.3
TDDFT framework
Figure 5.3: Effective one-electron potential (a), and ground state electron density (b)
of the (R1 , R2 = 47.7 Å, R3 = 61 Å) cylindrical NM. Results are shown as function of
the radial coordinate. Black lines: the NM with core radius R1 = 31.8 Å and gap size
S = 15.9 Å; Red lines: the NM with core radius R1 = 43.5 Å and gap size S = 4.2 Å.
The Fermi level is shown with dashed blue line in the panel (a).
79
Chapter 5
Figure 5.4: Frequency dependent absorption cross section per unit length for the
infinite cylinder with radius = 61 Å. The incident plane wave is linearly polarised
with electric field vector perpendicular to the cylinder axis. The quantum TDDFT
calculations (red line) are compared with classical calculations (black dashed line).
Classical theory uses Drude model for the dielectric function of Na with parameters
adjusted to TDDFT result.
with radius R = 40.96 a0 (21.7 Å) the maximum of the dipolar plasmon resonance in
absorption spectra is reached at the frequency ωsph = 2.98 eV. This is lower than the
√
class = ω / 3 = 3.4 eV.
frequency of the quasistatic Mie resonance for the sphere ωsph
p
√
Here ωp = 4πn+ = 5.89 eV is the Drude plasma frequency of Na. Similarly, for the
individual cylinder with radius R = R3 = 115 a0 (61 Å) the dipolar plasmon resonance
in absorption spectra shown in Fig. 5.4 appears at ωcyll = 4.02 eV which is lower than
√
class = ω / 2 = 4.16 eV.
the classical value ωcyll
p
5.3.1.
Tunneling effects
To study the quantum tunneling effects in this canonical structures, the TDDFT calculations are compared with classical electromagnetic theory where the dielectric function
of Na is described within the Drude model (DM), given by Eq. 1.2. The attenuation
and the plasma frequency are, respectively, η = 0.218 eV and ωp = 5.16 eV. The latter
80
5.3
TDDFT framework
is lower than the nominal Drude frequency 5.89 eV for Na metal. This choice of parameters allows to account for the spill out effects and plasmon line broadening because
of the decay into electron-hole one-particle excitations [196]. The good agreement with
TDDFT calculations for the individual nanowire is then reached as shown in Fig. 5.4
(similar agreement is reached for the individual nanosphere).
Provided good agreement between benchmark results for the individual nanoobjects,
the main differences between the TDDFT and classical Drude calculations for plasmonic
dimer or core-shell NM can be then attributed to electron tunneling. For the narrow
gaps the electron tunneling strongly modifies optical response of the core shell NM and
SD as follows from Fig. 5.5. In the panel a) of the Figure we show the absorption cross
section per unit length of the cylindrical NM. The frequency range corresponds to the
−
low energy bonding hybridized resonance ω−
. When the width of the gap reduces below
5.5 Å the tunneling current becomes strong enough leading to partial neutralization of
the plasmon induced charges across the gap. As the gap width reduces, the tunneling
−
efficiency grows, and the ω−
resonance broadens and progressively disappears. For the
gap size S = 4 Å the corresponding plasmon mode is quenched. Similar behaviour is
calculated for the bonding dipole plasmon of the spherical dimer. For the gap size below
4 Å the plasmon resonance in the absorption cross section becomes broadened by the
tunneling effects, and it basically disappears for S = 2 Å. Notably, as compared to the
NM, the smaller gap size is needed in the case of the spherical dimer to reach the same
tunneling efficiency. Indeed, the NM is characterised by the extended contact between
the core and the shell where the tunneling current flows transversally through the entire
core-shell gap. This is while in the case of the spherical dimer the tunneling current is
confined to the narrow contact region around the symmetry axis of the dimer.
5.3.2.
TDDFT approach for the systems under applied bias.
To achieve the control of the plasmon response while keeping the applied bias low, we
reduce the gap size down to 6 Å where tunneling effects, albeit weak, do appear. The
optical response of the ground state systems and that of the system under bias are
obtained from the quantum mechanical Time Dependent Density Functional Theory
(TDDFT)[188] calculations. The presence of the applied bias implies that the DC current flows through the gap. This situation represents a real challenge for the TDDFT
81
Chapter 5
Figure 5.5: Absorption spectra calculated with TDDFT for cylindrical (R1 , R2 =
47.7 Å, R3 = 61 Å) NM [panel a)], and R = 21.7 Å spherical dimer [panel b)]. The
incident plane wave is linearly polarised with electric field vector perpendicular to the
NM axis [panel a)], and parallel to the dimer axis connecting nanoparticle centers [panel
b)]. Results are shown for different sizes of plasmonic gap as indicated in the insets.
82
5.3
TDDFT framework
calculations performed here for a finite size system with a fixed number of electrons. Several methods have been proposed in literature [197–199] in order to obtain the transitory
steady state regime needed for the calculation of the absorption spectra. We used two
approaches and concluded on their near equivalency. (i) The first approach one consists
in calculating the ground state of the system in the presence of an external potential.
The system is polarized by the charge transfer between nanoparticles, and removal of
the external potential triggers the tunneling current. (ii) The second approach consist
in applying bias directly to the ground state of the non-polarized system.
In the case of the core shell NM we used the following strategy. First, we used the
stationary DFT to calculate the KS orbitals and an electron density of the system under
b .
the action of the time-independent external potential Vext
o
n
b
(r; n) ψjb (r)
T + Veff
=
n(r)
=
Ej ψjb (r),
2
X b
ψ
(r)
j (5.6)
j∈occ
b = V b + V b + V b , and the external potential is given by
where Veff
xc
ext
H
b
Vext
= V0 , ρ < R1 ,
ln(ρ/R2 )
b
, R1 ≤ ρ ≤ R2 ,
Vext
= V0
ln(R1 /R2 )
b
Vext
= 0, R2 < ρ,
(5.7)
here (ρ, φ, z) are the cylindrical coordinates. Superscript b indicates that the DFT
calculations were performed in the presence of an external time-independent potential.
The bias is applied to the core, and the shell is assumed to be grounded. For V0 > 0
the polarisation of the system is linked with electron transfer from the core to the
shell. The induced density screens the applied DC field so that the final total selfconsistent potential is very close to that of the ground state system without an external
b ≈ V 0 as we illustrate in Fig. 5.6 a). This is because the electron density
perturbation Veff
eff
of the system is changed only slightly by the applied potential. Indeed, the potential
of the charged cylinder is VQ = 2Q ln(ρ), where Q is the charge per unit length. Then
for the (78 a0 , 90 a0 , 115 a0 ) NM geometry [(41.3 Å, 47.7 Å, 61 Å)] the transfer of one
electron (out of 285) per 1 a0 length results in the 7.8 eV bias between the core and the
shell. Thus, as a consequence of the screening we have:
83
Chapter 5
b
Figure 5.6: a) Effective one-electron potentials Veff
obtained from the self-consistent
DFT calculations for the (41.3 Å, 47.7 Å, 61 Å) NM under applied external potential
given by Eq. (5.7). Calculations were performed for different amplitudes V0 of the applied external potential as explained in the insert. b) Hartree and exchange-correlation
b
b
contribution VHb + Vxc
to the total Veff
potential calculated for different magnitude V0
of the applied external potential as explained in the insert. Dashed lines show position
of the Fermi level of the system.
84
5.3
TDDFT framework
b
0
b
VHb + Vxc
≈ Veff
− Vext
,
(5.8)
0,
where the effective potential of the ground state system without an applied bias, Veff
is given by Eq. (5.5). We obtain that the sum of the Hartree and exchange-correlation
b recontributions to the self-consistent potential calculated in the presence of the Vext
b potential is applied to the ground state system.
produces the situation where the −Vext
This corresponds to the V0 bias applied to the core. Indeed, the V0 bias results in the
ēV0 = −V0 potential acting on the core electrons (ē = −1 is the electron charge). The
effective one-electron potential obtained with different magnitude V0 of the external potential is shown in Fig. 5.6 b). It is worth noting that inspection of the figure clearly
shows that the tunneling barrier between the core and the shell is reduced by applied
bias. For V0 above 4 eV, the electrons at the Fermi level of the shell can escape into the
core via an efficient over-the-barrier transition. For the reasons, explained below we will
denote V0 as targeted bias.
b given by
Once the self-consistent solution of Eq. (5.6) with external potential Vext
Eq. (5.7) has been obtained with DFT, the stationary KS orbitals ψjb (r) are used as
initial states for the second, TDDFT step of the study. The time dependent KS equations Eq. (5.2) and Eq. (5.3) are solved with time-dependent potential
b
Vext (t) = Vext
exp −(2.5t/T0 )2 ,
(5.9)
b is given by Eq. (5.7). Starting from t = 0, the external potential V b is prowhere Vext
ext
gressively removed and becomes nearly zero at T0 = 7.5 fs. For t > T0 the dynamics of
the density is driven by the time-dependent Hartree and exchange-correlation potentials,
where the initial polarisation of the system produces the effect of the V0 bias applied
to the core (see Fig. 5.6 b)). The smooth in time switching off function given by the
exp −(2.5t/T0 )2 is needed in order to avoid excitation of the electron density oscillations in the system. In Fig. 5.7 a) and b) we show the time evolution of the effective bias
Ueff (t) and tunneling current density J(t) through the middle of the S = 6.4 Å wide gap
for the (41.3 Å, 47.7 Å, 61 Å) NM. The effective bias Ueff (t) is a function of time and it
is defined as the difference between the one-electron potentials at the middle of the shell
and at the center of the core, respectively. It is in general different from the targeted
bias V0 imposed in the static DFT calculations. The positive current corresponds to the
electron transfer from the shell to the core.
85
Chapter 5
Figure 5.7: TDDFT calculations for the electron dynamics of the (41.3 Å, 47.7 Å,
b
triggers the
61 Å) cylindrical NM. Progressive removal of the external potential Vext
time evolution of the effective bias Ueff (t) [panels a) and c)], and density J(t) of the
tunneling current through the middle of the core-shell gap [panels b) and d)]. Results
are shown for different amplitude of the targeted bias V0 as explained in the inserts.
Panels a) and b) show the calculations where no stabilization of the effective bias has
been used. Panels c) and d) correspond to the stabilized case (further details are given
in the text).
b ≈ 0, and both U (t) and J(t) reach maximum. Observe that
At t = T0 the Vext
eff
the effective bias Ueff (T0 ) is lower than V0 . This is because within the time interval
0 ≤ t ≤ T0 the initial charge separation between the core and the shell has been partially
neutralized by the tunneling current through the gap. At larger times the polarisation
of the system is further reduced because of the tunneling current, and Ueff (t) decreases
reflecting equilibration of the Fermi levels of the core and the shell. The latter, in turn,
leads to the reduction of J(t). Considering that in the polarised system the charge
of the core is Q(t), the potential difference between the core and the shell is given by
∆Φ = 2Q ln [R2 /R1 ]. Since the current I(t) = −dQ(t)/dt we can write:
dUeff
d∆Φ
R1 + R2
≈
= −2π
J(t) ln [R2 /R1 ] .
dt
dt
2
(5.10)
The current in its turn depends on the applied bias via J = GUeff , where G = dJ/dU is
the conductance of the junction. The low bias conductance G is independent of U , and
86
5.3
TDDFT framework
the linear dependence of J (and of I) on Ueff leads to the solution of Eq. (5.10) given
by exponential decay of the initial polarisation.
For the effective bias values below 1 eV the tunneling current is low, and the charge
separation between the core and the shell can be maintained for a relatively long time.
Therefore, we obtain quasi steady-state situation with nearly constant tunneling current
over large time interval (∼ 10 - 15 fs). For the higher V0 (and Ueff ) the tunneling current
is strongly increased. Because of the large tunneling current the polarization of the
system appreciably varies with time for t > T0 and so does the tunneling current. No
steady state regime is reached and the calculation of the optical response becomes not
an easy task in this case. Additionally, the current density J(t) features an oscillating
structure at large times as seen in Fig. 5.7 b). It can be attributed to the excitation
of the collective plasmon modes due to the finite spectral width of the switching function.
Despite the difficulty to reach the steady state, we nonetheless have applied impulsive
perturbation at t = T0 , and the frequency ω-dependent optical absorption cross section
σ(ω) has been obtained as is the standard ground state TDDFT calculations. These
results can be tentatively interpreted as corresponding to some average value of the bias
within the T0 ≤ t . T0 + 15 fs time interval determining convergence of the time-tofrequency Fourier transform.
To reach, at least transitory, the steady current even at relatively high bias we have
performed the calculations where an effective bias Ueff (t) is stabilized at constant level
U for t > T0 . This is achieved by adding the correcting potential of the form given by
Eq. (5.7) with an amplitude calculated dynamically so that Ueff (t) = Ueff (T0 ) ≡ U, t >
T0 . In Fig. 5.7 c) and d) we show the corresponding results for the effective bias and
current, respectively. With the present choice of the parameters of the system, for V0
below 4 eV (Ueff (t) below ∼2.5 eV) the quasi steady-state is reached within the time
interval T0 ≤ t . T0 + 15 fs allowing calculation of the optical properties σ(ω) with
impulsive perturbation. The relation between the targeted bias V0 and the effectively
reached steady state bias U can be deduced from Fig. 5.7 c), where approximate relation
U ∼ V0 /2 holds in overall. The steady state is transitory because at large times too
much of the charge is transferred between the core and the shell so that the artificial
potential correction becomes too large leading to unphysical variations of the current.
87
Chapter 5
Figure 5.8: Absorption cross section per unit length of the (41.3 Å, 47.7 Å, 61 Å)
cylindrical NM obtained from the TDDFT calculations with (Stabilized) and without
(Non Stabilized) stabilization of the tunneling current and effective bias. We also show
results of the classical electromagnetic calculations where the tunneling is accounted
for with quantum corrected model QCM. Red dots are used in all panels to trace
the reference result corresponding to the ground state system without any external
potential. Colors are used to distinguish results obtained at different amplitude V0 of
the polarising potential (targeted bias). The relation between an effective stabilized
bias U (first column) and the targeted bias V0 (second column) is given in the inset of
the panel with QCM results.
88
5.4
5.4.
Control with applied bias.
Control with applied bias.
In Fig. 5.8 we show the absorption spectra calculated with TDDFT for the (41.3 Å,
47.7 Å, 61 Å) cylindrical NM with the width of the core-shell gap reduced to S = 6.4 Å.
The incident plane wave is polarized perpendicular to the symmetry axis of the cylindrical NM. Red dots show the reference result: absorption cross section of the ground state
system without an applied bias. The frequency range covered by the data corresponds
−
to the lowest energy bonding hybridized plasmon resonance ω−
which is most sensitive
to the variations of the tunneling barrier. Different curves are labelled according to the
targeted bias V0 . Indeed, for the non stabilized case the effective bias between the core
and the shell varies upon σ(ω) calculation and thus it is ill-defined. The insert of the
lowest panel of the figure showing the results obtained with quantum corrected model
(discussed below) allows to connect V0 and the bias U for which the σ(ω) is obtained in
the stabilized TDDFT calculations.
As a main result of this thesis, the applied bias leads to the broadening of the plasmon
resonance in absorption cross section, and to the reduction of the resonant absorption
(see Fig. 5.8). The plasmon mode slightly blue shifts with increasing bias. These results
can be explained as due to the bias induced reduction of the tunneling barrier so that the
tunneling current increases and thus (i) resistive losses to the electron hole excitations
are larger; (ii) capacitive coupling between the core and the shell is reduced because
of the partial neutralisation of the plasmon-induced screening charges across the gap.
These qualitative trends are robust and present in both (stabilized and non stabilized)
TDDFT data showing 30-35 % variation of the maximum absorption for only some eV
bias change. The variation is somewhat larger in the stabilised case. This is because, for
the same initial V0 , it corresponds in average to the larger effective bias and so to the
larger conductivity of the junction for t > T0 time interval relevant for the calculation
of σ(ω) (see Fig. 5.7).
To gain insight into the main physics underlying the TDDFT results let us consider a
canonical system formed by a junction between two flat metal surfaces separated by
a narrow gap of width S. We can use a quantum many-body approach developed for
electron dynamics in metal-isulator-metal tunnel junctions in the presence of electromagnetic fields to describe linear and non-linear effects such as rectification[200]. Within
the linear response, the presence in the junction of the optical field Eω at frequency ω,
89
Chapter 5
triggers a tunneling current at the same frequency. The amplitude of the dissipative
component (in phase with the driving field) of this ac current density is given by:
Jω (U, Vω ) =
Vω
[Jdc (U + ω) − Jdc (U − ω)] ,
2ω
(5.11)
where U is the applied dc bias, Vω ' SEω is the optical bias in the junction, and
Jdc (U ± ω) is the dc current at bias U ± ω. For slow variation of Jdc with U , we obtain
the classical limit
Jω (U, Vω ) = S
dJdc (U )
Eω ,
dU
(5.12)
which can be deduced by developing J (U + SEω cos(ωt)) in the Taylor series for SEω U . An increase of the applied bias leads to a lower tunneling barrier, thus the conductivity σ(U, S) = S dJdc /dU becomes larger resulting in a larger tunneling current at
optical frequency Jω . In the cases, where an increase of the tunneling current has been
caused by the progressive reduction of the gap width S (with no applied bias), it was
responsible for the attenuation of the gap plasmon resonance[38, 39, 42]. In the situation
like here where a bias is applied (with fixed gap separation), the increasing tunneling
current equally leads to a smaller absorption cross section at the plasmon gap resonance.
The screening charges at the opposite sides of the junction are partially neutralised and
the resonance is broadened as a result of the increased resistive losses. It is worth to
mention that for larger gap size S, the tunneling probability decreases so that larger
applied bias U is needed to reach the same tunneling current, and thus a similar degree
of plasmon response modification.
5.5.
Quantum corrected model. QCM.
To further verify the validity of the concept of bias-assisted tunneling at optical frequencies, we have also calculated the optical response of the cylindrical NM using the
quantum corrected model (QCM)[1]. The permittivity of the Na core and Na shell is
described with Drude model given by Eq. (1.2) with parameters adjusted to the TDDFT
result for individual nanowire (nanosphere). The tunneling gap is described using a local
effective dielectric function
εg (U, S) = 1 + i
90
4πσg (U, S)
,
ω
(5.13)
5.5
Quantum corrected model. QCM.
which includes the effect of the conductivity across the gap at optical frequencies, in such
a way that the ”quantum” relationship between the tunneling current and the optical
field in the gap is correctly reproduced. The constant size of the gap S in this case
allows to use the TDDFT results to define the parameters of the effective medium in the
presence of the bias. To this end, within the simple approach widely applied to describe
the optical rectification and frequency mixing in tunneling junctions, the total current
density through the junction reads[89, 200]:
Jtot = Jdc (U, S) + Jopt = Jdc (U, S) +
dJdc (U, S)
Uopt (ω) e−iωt .
dU
(5.14)
Here Jdc (U, S) is the dc current due to applied bias U , Uopt (ω) e−iωt is the optical bias
induced by electromagnetic field in the junction. The quantity G(U, S) =
dJdc (U,S)
dU
is
the dc conductance of the junction. Since for the QCM model one is interested in the
conductivity of the gap σg we recast Eq. (5.14) in the form:
Jtot = Jdc (U, S) + Jopt = Jdc (U, S) +
dJdc (U, S) dU
Eopt (ω) e−iωt ,
dU
dE
(5.15)
where E = U/S is the radial component of the dc field in the gap because of the applied
bias U , and Eopt (ω) is the optical field in the gap. The quantity
σg (U, S) = S
dJdc (U, S)
,
dU
(5.16)
is then the sought conductivity of the gap.
The current-voltage characteristic of the junction is presented in Fig. 5.9 for the (41.3 Å,
47.7 Å, 61 Å) cylindrical NM discussed in the text. J(U, S) is obtained from the stabilized current and effective bias values reported in panels c) and d) of Fig. 5.7. The
red curve shows an analytical fit to the TDDFT data. It is given by the equation
J(U, S) = 0.00332U e0.548 U , where U is measured in eV and current density in nA/Å2 .
The analytical fit greatly simplifies calculation of the differential conductance of the
junction dJ/dU , and extrapolation to large U - values, where the TDDFT calculations
are not available.
The QCM results obtained with parametrisation of the tunneling gap as described above
and with no further parameter adjustment are shown in the lowest panel of Fig. 5.8. The
good quantitative agreement of plasmon peaks intensity and broadening when compared
91
Chapter 5
Figure 5.9: Dependence of the tunneling current density on the applied bias for the
(41.3 Å, 47.7 Å, 61 Å) cylindrical NM. Dots stand for the TDDFT data obtained from
panels c) and d) of Fig 5.7, and solid line is an analytical fit with equation indicated in
the insert.
with the TDDFT results indicates that the bias-induced variation of the conductivity
across the junction is at the origin of the change of the absorption cross section. Indeed,
applied bias can produce various changes in the system which might lead to the modification of the plasmon modes: (i) Change of the potential well ensuring the electron
confinement; (ii) Change of the non-local screening because of the change of electron
density at facing surfaces across the gap; (iii) Change of the tunneling barrier and thus
of the tunneling current across the junction. The QCM accounts only for the last effect,
and it appears sufficient to describe the TDDFT data.
Using QCM we could also calculate absorption spectra for the bias U as high as 3, 4,
and 5 eV which is technically impossible with present full quantum TDDFT approach.
The tunneling current between the core and the shell is too strong in this case leading
to the fast charge equilibration in the system so that no steady state can be reached.
The QCM results in lowest panel of Fig. 5.8 predict quenching of the plasmon resonance
for the applied bias above 4 eV.
92
5.6
5.6.
Time dependence of the induced dipole
Time dependence of the induced dipole
The difficulty to obtain the steady state regime for the extraction of the absorption
spectra calls for consistency checks in order to provide further support for the interpretation of the TDDFT results. Broadening of the plasmon resonance in optical response
with increasing bias indicates an increased plasmon coupling to the electron-hole binary
excitations because of the larger tunneling probability. In terms of electrical engineering,
the higher barrier conductivity leads to larger resistive losses[200–202]. The broadening of the resonance peak in the frequency domain corresponds, in the time domain,
to faster coherence loss and decay of the plasmon mode. To illustrate the dynamics of
the plasmon resonance we show in Fig. 5.10 a) the time evolution of the induced dipole
excited in the (41.3 Å, 47.7 Å, 61 Å) cylindrical NM by the incident infra-red pulse. The
NM is initially in the ground state. No bias is applied. The incident pulse is polarised
perpendicular to the symmetry axis and its electric field is given by:
" #
t − 3TIR 2
,
EIR (t) = E0 cos(ωIR ) exp −
TIR
(5.17)
−
with ωIR = 0.82 eV at resonance with bonding hybridized plasmon ω−
of the NM, and
duration of the pulse 2TIR = 20 fs. The EIR (t) is shown with red dashed line in Fig. 5.10
a). Essentially, the incident field excites the plasmon mode and the corresponding density
oscillation persist long time after the termination of the incident pulse. This is illustrated
with black line showing result obtained for the isolated NM in vacuum in absence of the
applied bias. One can observe slight Rabi oscillations for t > 80 fs originating from the
interference between plasmon and electron hole excitations[126].
At the time T0 = 59 fs the bias is switched on, i.e. the external potential expressed by
Eq. (5.7) with negative V0 is applied to the system, resulting initially in the effective bias
U = −V0 as shown on Fig. 5.10 b). The suddenly applied bias leads to the population and
coherence decay of the collective plasmon oscillations as manifested from the comparison
with U = 0 result (black line) in Fig. 5.10 a). This is fully consistent with broadening
−
and loss of intensity of the ω−
resonance in absorption spectra. Because of the electron
tunneling and polarisation of the system, the applied field is progressively screened, and
the potential energy difference between the core and the shell decays in time. For the
highest applied bias the tunneling current is strongest, and U (t) quickly relaxes from
initial value of 5 eV to much lower value of ∼ 2.5 − 3 eV. For these lower values of
effective bias the tunneling current is relatively small. The energy alignment of the
93
Chapter 5
Figure 5.10: The time evolution of the induced dipole P (t) [panel a)] and effective
bias U (t) [panel b)] calculated with TDDFT for the (41.3 Å, 47.7 Å, 61 Å) cylindrical
NM. Initially the non-perturbed ground state system is subjected to the IR pulse res−
onant with ω−
plasmon mode and polarized perpendicular to the NM axis. After the
termination of the pulse, at T0 = 59 fs the positive bias U is applied to the core while
the shell is grounded. Different curves correspond to the results obtained with different
U -values as explained in the insert.
94
5.7
Summary of main results and conclusions
40
TDDFT
TDDFT
U
0.2
a)
s (nm2)
s/l (nm)
0.4
U
0
0.5
0.7
0.9
w (eV)
1.1
b)
0
2.2
2.6
3.0
w (eV)
3.4
3.8
QCM
0.6
0.4
dipole
s/l (nm)
20
0
0.2
0
c)
0.5
0.7
0.9
w (eV)
1.1
d)
0
5
10
15
20
time (fs)
Figure 5.11: Effect of an applied bias on the plasmonic modes of the cylindrical NM
and the plasmonic dimer. Panels a), b): TDDFT results for the absorption spectra
of the (41.3 Å, 47.7 Å, 61 Å) cylindrical NM and of the plasmonic dimer formed by
R = 21.7 Å spherical nanoparticles separated by a gap of width S = 6.4 Å. For the
NM, the absorption cross section per unit length, σ/l, calculated with TDDFT, is
compared with the results of the classical calculations within the quantum corrected
model (QCM) shown in panel c). Panel d): time evolution of the dipole induced in
−
the NM of panel (a) by an incident pulse of light resonant with ω−
plasmon mode,
and linearly polarised in the direction perpendicular to the symmetry axis. Long times
following the termination of the incident electromagnetic pulse are shown. At t = 2 fs
the bias is applied to the system. In each panel the calculations have been performed
for different values of the applied bias, as detailed in the inserts.
Fermi levels of the core and the shell with exponential decay of U (t) proceeds on longer
time scales. Oscillating structure of the U (t) curve is caused by the excitations in the
system produced by the sudden change of the potential at t = T0 . In particular, the
+
higher energy ω−
plasmon mode with core character[4] is excited.
5.7.
In order to get a crystal clear idea of the proposed active control strategy with an applied
bias we summarize in Fig. 5.11 the main results obtained in this chapter. In Fig. 5.11(a)
95
Chapter 5
we show the absorption spectra calculated with TDDFT for a cylindrical NM with the
width of the core-shell gap of S = 6.4 Å. The incident plane wave is polarized perpendicular to the symmetry axis of the cylindrical NM. For the sake of completeness
we show in Fig. 5.11(b) results obtained by the Orsay team (D.C. Marinica and A. G.
Borisov) for the absorption spectra for a spherical plasmonic dimer with the same gap
width (S = 6.4 Å). The incident plane wave is polarized along the symmetry axis of
the dimer. The calculations have been performed for different values of the stabilised
bias U applied between the core and the shell of the cylindrical NM or between the
nanoparticles forming the dimer as depicted in the insets of Fig. 5.11(a) and (b).
When the bias across the gap is applied, a considerable decrease of the resonance
absorption peak is produced for both geometries. This is the central result of this
chapter. Within the studied U range 0 < U < 3 eV we calculate ∼30 % variation of
the maximum of the absorption cross section primarily because of the broadening
of the plasmonic resonance. Thus, even a moderate change of the DC bias applied
allows to appreciably modify the intensity of the plasmon response.
The TDDFT results are retrieved with classical electromagnetic calculations using
QCM as shown in Fig. 5.11(c) for cylindrical NM. Since tunneling is the only effect
accounted for in QCM, good agreement between the TDDFT and QCM results
indicate that this is indeed the main physical mechanism behind the bias induced
change in plasmon responce. At high applied bias, where the TDDFT calculations
−
can not be performed, the QCM predicts the quenching of the ω−
resonance for
U above 4 eV.
For the bonding dipole resonance of the spherical dimer as shown in Fig. 5.11(b),
the peak in absorption cross section notably blue shifts with increasing U. This
result is consistent with the reduction of the capacitive coupling across the gap
because of the neutralization of the plasmon induced screening charges as has been
discussed in the case of molecular shunted junctions[159].
The dependence of the absorption cross section on the applied bias opens the path to
controllable modification of the plasmon dynamics in the system. This is illustrated in
Fig. 5.11(d) where we show the time evolution of the dipole induced in the core shell
cylindrical gap by the external Gaussian electromagnetic pulse resonant with the gap
plasmon mode. After the termination of the pulse, the time evolution of the induced
96
5.7
−
dipole is given by the decaying oscillations at the plasmon frequency ω−
, as shown with
a black line in the figure for the case in the absence of bias. We compare this reference
result with the calculations of the response when the bias between the core and the shell
has been switched ”on” at t = 2 fs. The larger the applied bias is, the faster the decay
and dephasing of the collective plasmon oscillation in the system due to the increase
of the electron tunneling. This result, monitored in the time domain, is equivalent to
the broadening of the absorption cross section when the bias is applied, observed in the
frequency domain response of panels (a), (b), and (c) of Fig. 5.11.
In conclusion, we have proposed a novel strategy for active control of junction plasmon
resonances based on the application of a bias across the gap. The physical origin of the
effect is a bias-induced change of the electron tunneling barrier which in turn controls
the conductive coupling between the two nanostructures. The feasibility of the approach
has been demonstrated with proof-of-principle calculations based on the quantum mechanical time-dependent density functional theory. While we here considered vacuum
gaps, filling the plasmonic gap with dielectric materials such as oxides will also modify
the tunneling barrier. This effect may be useful in practical realizations of the proposed device since it introduces additional tuning modalities. Thus, the lowering of the
tunneling barrier offers the possibility to reach the sought control for broader gaps. Further extensions of this concept may include molecular linkers with conductance windows
allowing for ’on’/’off’ switch functionalities.
Quantum active control of plasmons, as demonstrated here, is inherently a fast (ps)
process allowing operation at the time scales of modern electronics, reversible, and progressive tuning of the plasmon resonances. This opens appealing perspectives for the
development of tunable absorbers for solar energy harvesting, control of the information transfer in plasmonic waveguides, and manipulation of plasmon-exciton couplings.
Our concept of electrical control of light in metallic nanostructures thus provides a new
platform for many practical applications in photonics and optoelectronics.
97
Chapter 6
Quantum Plasmonics and charged
clusters dynamics
In this chapter, using the TDDFT we performed full quantum calculations of the optical
response of the charged spherical clusters. We have obtained that, because the system
stays neutral in the bulk, and the extra-charge is localized at the surface of the cluster,
the dipolar plasmon mode displays only small frequency shift linked with change of
the electron spill-out from the nanoparticle boundaries [7]. For the negatively charged
clusters, we have also addressed the issue of the decay of the negative charge via resonant
electron transfer from the cluster orbitals into the continuum of propagating states
above vacuum level. Indeed, even small (relative to the total number of electrons in
the system) negative charge raises the Fermi level of the cluster above the vacuum level
and renders the system unstable. The excess of the electron population decays with
characteristic time constants that can be fully understood using an analytical study
based on the WKB method, similar to the studies of the alpha-decay. Our results on
the charged clusters allow to critically reinterpret the experimental data obtained with
electrochemistry, where the shift of the plasmon resonances has been attributed to the
charging effects. Based on our results, we tentatively propose that the frequency shift
of plasmon resonances is related with change of the dielectric environment in vicinity of
the cluster.
99
Chapter 6
6.1.
Quantum Plasmonics and charged clusters dynamics
Introduction
As we have already discussed in the introductory part of Chapter. 5, there is a continuous effort in the plasmonic community to achieve a control of the optical response
of the nanostructures not only via the proper design of these nanostructures [203, 204],
but actively. For example, by electrically driving liquid crystals containing plasmonic
particles [205]. The modulation of the frequencies of plasmon resonances has been also
achieved in electrochemistry appliying bias between the nanoparticles and electrolyte or
by adding chemical reductants to a colloidal nanoparticle solution [179, 180, 206–210].
In some cases the plasmon wavelength shift of 10 nm per eV of applied potential has been
reported [211]. By analogy with 2D planar systems such as graphene [53, 178, 212–214]
it is very tempting to explain the observed shift of the plasmon frequency as a result of
the charge doping
q of the nanoparticles [206]. Indeed, since the bulk plasmon frequency is
given by ωp =
ne2
ε0 m ,
is at frequency ωDP
and e.g. dipolar plasmon resonance for the spherical nanoparticle
√
= ωp / 3 one would expect that the change in the conduction elec-
tron density ∆n would automatically lead to the frequency shift of the localised plasmon
resonance with ∆ωp /ωp ∼ ∆n/n. However, the situation is much more complex than it
seems at a first glance. (i) Because of the high density of the conduction electrons in
metal ∆n should be also large which implies strong charging of the system resulting in
high electrostatic potentials [178]. (ii) The negative charge Q of the cluster of radius
R leads to the electrostatic potential |Q|/R acting on the electrons in the cluster, so
that the Fermi energy EF of the system is rased by the same amount, which can lead
to the electron loss into the continuum of propagating states above vacuum level for
|Q|/R > Φ, where Φ is the work function of the nanostructure. (iii) In difference to the
2D planar structures, in virtue of Gauss theorem, in the 3D metallic objects the volume
stays neutral. The screening charges reside at the surface within the narrow layer of
the width given by the screening length rs (Wigner-Zeitz radius of the metal). It is not
obvious that such a narrow layer of surface charge might lead to the“expected” shift the
plasmon frequency.
The main goal of this chapter is to study the electron dynamics and plasmon modes of
small charged metallic clusters, where using the TDDFT calculations allows to reveal the
role of different effects discussed above in the optical response of these nanostructures.
Below we describe spherical sodium and silver clusters with sizes ∼ 2 nm as has been
100
6.2
Model and computational aspects
used in this study, and present the ground state properties of these neutral and charged
systems as obtained in the density functional theory calculations (DFT). The static DFT
results provide an input for the TDDFT study of the electron dynamics and optical
response as function of the cluster charge. As well we use TDDFT to address the
electron losses for the negatively charged clusters. Finally, we conclude underlying the
consequences that our results can have for the interpretation of the electrochemical
experiments.
6.2.
Model and computational aspects
The sodium and silver spherical nanoparticles studied here are described using jellium
metal (JM) approximation, where the positive ion cores are not treated explicitly, but
represented with uniform positive background charge of the density n+ = (4πrs3 /3)−1 .
The Wigner-Seitz radius (or the screening length of the free electron gas) rs equals to 4 a0
for sodium, and 3.02 a0 for silver1 . In the ground state (neutral system) both clusters
have a closed shell structure and contain 2018 valence electrons, so that the positive
background charge is also Q+ = 2018 which results in the cluster radius R = 50.6 a0
(2.67 nm) for sodium, and R = 38.2 a0 (2.06 nm) for silver. Despite its simplicity, the
JM correctly captures the collective plasmon modes of conduction electrons, and it has
demonstrated a good predictive power in the description of quantum effects in individual
nanoparticles and in narrow gaps [54, 55, 122]. For sodium, which is the prototype
of the free electron metal, the JM performs particularly well in the description of the
interaction of the optical pulse with nanosized objects. For silver, the contribution of the
localized d-electrons to the screening has to be taken into account in order to correctly
reproduce frequencies of the localised plasmon mode(s) [127, 128]. This is achieved via
the introduction of a polarizable background [38] characterised by the non-dispersive
dielectric constant ε∞ = 4.58. For the neutral sodium cluster we calculate the work
function ΦN a = 3 eV in good agreement with earlier studies on such systems [215–217].
For the silver cluster, in order to reproduce the experimentally measured work function
of silver ΦAg = 4.65 eV [218], an additional attractive potential of 3 eV is imposed inside
cluster (the so-called stabilized jellium model [219]).
Prior to the TDDFT studies of the electron dynamics in the neutral and charged clusters, their ground state properties have been determined with static density functional
theory DFT calculations. The Kohn-Sham scheme of the DFT is used [220], where
1
the Bohr radius a0 = 0.053 nm
101
Chapter 6
the electron density is given by that of the non-interacting system n(~r) =
P
j
|ψj (~r)|2 .
The summation runs over all occupied Kohn-Sham orbitals ψj (~r), and ψj (~r) obey the
stationary Schrödinger equations
(T + Vxc (~r; n) + VH (~r; n) + Vst (~r)) ψj (~r) = Ej ψj (~r).
(6.1)
In Eq. (6.1), T is the kinetic energy operator, Vxc (~r; n) is the exchange-correlation potential derived within the local density (LDA) from the exchange-correlation functional
of Gunnarson and Lundqvist [130], and VH (~r; n) is the Hartree potential calculated from
the charge density N+ − n. The exchange-correlation and Hartree potentials are equivalent to those introduced in the context of the TDDFT with only difference that the
time-dependence is absent in the static calculations. Finally, Vst (~r) is the stabilizing
potential used only in case of Ag clusters.
6.3.
Ground state properties
In Fig. 6.1 we show the ground state electronic density calculated with DFT for the
charged Na and Ag clusters. Results are presented for different values of the cluster
charge Q. Within the range of the cluster charges considered here, with exception of
the largest positive cluster charge Q = +20, the valence electron density is not strongly
affected by the charging. Generally it is close to 1 (in units of the background positive charge density n+ ), and features Friedel oscillations because of the reflection of the
electron wave at nanoparticle boundaries. One also clearly observes the spill out of the
electron density outside cluster boundaries with spatial extension of this effect slightly
larger for Na because of the lower work function and smaller electron confinement in the
potential well of the nanoparticle.
To reveal the actual location of the extra charges we analyse in Fig. 6.2 the change
of the ground-state electron density ∆n(r) = nQ (r) − n0 (r) induced by the charge Q
added to the cluster. Here nQ (r) is the electron density of the cluster with charge Q(r),
n0 (r) is the ground state electron density, and r is the radial coordinate. As particularly
obvious from panels e) and f) of this figure, the extra charge added to the cluster resides
mainly in the surface layer of the width given by the screening radius rs . Indeed, the
Rr
charge inside the sphere of radius r is given by 0 4πr02 ∆n(r0 )dr0 . It is almost zero for
102
6.3
Q=-40
Q=-30
Q=-20
Q=-10
Q=0
Q=+10
Na
Cluster
Q=+20
Ag
Cluster
Rcl
Rcl
Rcl=50.6 a.u.
Rcl=38.2 a.u.
a)
b)
Radial Coordinate (a. u.)
Figure 6.1: DFT results for the ground state electronic density of the neutral and
charged metallic clusters. The density (in units of the positive background charge
density n+ is shown as function of the radial coordinate measured from the center of
the cluster. Panel a) shows the results for sodium and panel b) for silver. The dashed
Ag
Na
vertical line, indicates the cluster edge (Rcl
= 50.6a.u.= 2.67nm and Rcl
= 38.9a.u.=
2.06nm ). Different colors are used to distinguish results obtained for different charge
of the clusters as explained at the top of each panel.
r < Rc , and it quickly reaches the nominal value of Q for r ' Rc + rs . This result
can be easily understood from the Gauss theorem in electrostatics. The excess charge
in some volume area of the cluster would lead to the electric fields through the surface
enclosing this volume area. In response to electric fields the conduction electrons which
are quasi-free inside the cluster, will move until the fields are screened, or equivalently
the excess charge is compensated. Thus, aside from the possible finite size effects the
electron density in the bulk stays largely unaffected by charging of the cluster and all
the excess charge is accumulated in the thin surface layer.
A charge Q located at the surface of the cluster of radius Rc , creates for the electrons
inside the cluster an additional constant potential V = −Q/Rc . Thus adding 20 electrons
to the Ag cluster already containing 2018 electrons (1 % change in the total number of
electrons), leads to the 14 eV potential for the electrons at the cluster surface. The
electrons outside the cluster and located at a radial distance r from its center experience
the −Q/r Coulomb potential. Based on these simple considerations we can make an
103
Chapter 6
Na Cluster
Δρ(Q=-40)
Δρ(Q=-30)
Δρ(Q=-20)
Δρ(Q=-10)
Δρ(Q=+10)
Δρ(Q=+20)
Δρ(Q=+30)
Ag Cluster
b)
a)
4π r2 Δρ(Q=-50)
4π r2 Δρ(Q=-40)
4π r2 Δρ (Q=-30)
4π r2Δρ(Q=-20)
4π r2 Δρ(Q=-10)
4π r2 Δρ(Q=+10)
4π r2 Δρ(Q=+20)
4π r2 Δρ
4π r2 Δρ
4π r2 Δρ(Q=-40)
4π r2 Δρ(Q=-30)
4π r2 Δρ (Q=-20)
4π r2Δρ(Q=-10)
4π r2 Δρ(Q=+10)
4π r2 Δρ(Q=+20)
4π r2 Δρ(Q=+30)
Na Cluster
Ag Cluster
d)
c)
(Q=-40)
(Q=-30)
(Q=-20)
(Q=-10)
(Q=+10)
(Q=+20)
(Q=+30)
Q(inside) (a. u.)
Q(inside) (a. u.)
Δρ (Q=-50)
Δρ(Q=-40)
Δρ(Q=-30)
Δρ(Q=-20)
Δρ(Q=-10)
Δρ(Q=+10)
Δρ(Q=+20)
Na Cluster
(Q=-50)
(Q=-40)
(Q=-30)
(Q=-20)
(Q=-10)
(Q=+10)
(Q=+20)
Ag Cluster
f)
e)
Figure 6.2: Induced density as function of the radial coordinate r for different values
of the cluster charge as calculated with DFT for sodium [panel a)] and silver [panel b)]
clusters. Panels c) and d): the charge induced at the surface of the sphere of radius r:
4πr2 ∆n(r) as a function of the radial coordinate for charged sodium c) and silver d)
clusters.
Panels e) (sodium) and f) (silver): the charge inside the sphere of radius r:
Rr
02
0
4πr
∆n(r
)dr0 as function of r.
0
estimate for the Fermi energy EF of the charged cluster as:
EFQ = −Φ − Q/Rc ,
104
(6.2)
6.3
Fermi Level (eV)
30
Silver
Sodium
20
10
0
-10
-20
-40
-20
0
Charge (a. u. )
20
Figure 6.3: Energy of the Fermi level EF (eV) as function of the cluster charge. The
DFT ground state calculations have been performed for Na (red circles) and Ag (black
circles) clusters a explained in the text. Straight lines of the corresponding color show
analytical dependence given by Eq. 6.2.
where Φ is the work function of the neutral cluster. Alternatively the work function is
given by
ΦQ = Φ + Q/Rc ,
(6.3)
i.e. negative charge decreases the work function of the nanoparticle, and, alternatively,
the positive charge increases the electron binding. For the affinity level of the singly
charged clusters one should rather use A = −Φ + 0.5/Rc as can be obtained considering
the work to add the extra charge −1 to the neutral system [215, 216]. For multicharged
clusters considering charge variation −|Q| → −|Q| − 1 this approach leads to Eq. 6.2.
To illustrate the effect of the cluster charge on the electron binding we show in Fig. 6.3
the Fermi energies of the clusters of different charge obtained in DFT ground state
calculations.
105
Chapter 6
The negatively charged cluster becomes unstable for the critical charge when EFQ ≥ 0,
i.e. when Q < Qcr given by
Qcr = −ΦRc .
(6.4)
When the number of the extra electrons added to the cluster exceeds |Qcr | (the cluster
charge is Qcr + q where q < 0), the Fermi level is promoted above the vacuum level.
The electronic population of the cluster decays via energy-conserving resonant electron
transfer into the continuum of vacuum states through the potential barrier separating
two classically allowed regions of motion: inside cluster r ≤ Rc and in vacuum Rext ≤ r
as we show in Fig. 6.4. Assuming that an electron far enough from the charged cluster
experiences the Coulomb potential −Q/r, and using Eq. 6.2 Rext can be obtained from:
EFQ = −Φ −
−Q
Q
=
,
Rc
Rext
(6.5)
so that
Rext =
Rc
Q
=
.
Φ + Q/Rc
1 − ΦRc /|Q|
(6.6)
As a result of the population decay, the cluster charge would relax to Qcr , where the
Fermi level is brought to the vacuum level and the system becomes stable. Taking
into account that the number of electrons already present in the cluster Ne is given by
Ne = [Rc /rs ]3 , the relative change of the charge allowed before the onset of the electron
population decay is given by:
|Qcr |/Ne = ΦRc / [Rc /rs ]3 ,
(6.7)
|Qcr |/Ne = Φ rs3 /Rc2 .
(6.8)
so that finally
Thus, the relative number of electrons that can be added to the neutral cluster before
the onset of the electron population decay decreases as Rc2 . The larger is the cluster, the
less electrons in relative terms it can admit. For the 2018 electron clusters studied here,
the Qcr is -5.6 for Na and -6.5 for Ag. This implies that the system becomes unstable
when only 6 (7) electrons are added to the Na(Ag) cluster which represents less then 0.4
% variation of the total number of electrons. We note in passing that, for the positively
charged clusters, accumulation of the non-screened ions at the cluster surface leads to
the so-called Coulomb explosion where the ions are ejected from the crystal lattice sites
into the vacuum [221–223]. Description of this process requires to consider the heavy
106
6.3
a)
b)
c)
Figure 6.4: a) Schematic representation of the effective one-electron potentials for a
small metallic negatively charged cluster. DFT result: red line. Pure Coulomb barrier
|Q|/r: blue line. Horizontal dashed line shows energy of the Fermi level modified by the
cluster charge. The dashed vertical lines, represent the limits of the classically allowed
regions of the electron motion (turning points), which are the cluster radius Rc and the
limit Rext given by the |Q|/r Coulomb barrier. For Rc < r < Rext an electron is in the
tunneling regime. Panels b) (Sodium) and c) (Silver) show the effective one-electron
potentials (continuous lines) and Fermi level energies (dashed lines) calculated with
DFT for the charged clusters as explained in the inserts.
107
Chapter 6
particle dynamics and has not been attempted here.
The fact that only limited number of electrons that can be added to the cluster before the
electron population starts to decay require some special treatments if one wants to look at
the ground state properties of the“overcharged” clusters. To obtain the results discussed
above for negative charge Q < Qcr we performed the DFT ground state calculations for
the restricted geometry, where the size of the computational mesh is set such that the
covered range of the radial coordinates r < Rext . r ≥ Rext corresponds to the region
of the classically allowed motion outside cluster for the electron with Fermi energy, as
shown in Fig. 6.4. This allows to restrain electrons to inside cluster and to converge the
ground state calculations even for the negative charges exceeding the threshold value.
6.4.
Absorption cross-section
In order to calculate the absorption cross-section of the charged clusters under the presence of an electromagnetic wave, we use the Kohn-Sham (KS) formulation of the time
dependent density functional theory [129] detailed in the previous Chapter. The ψj (~r)
orbitals obtained in the DFT ground state study are used as an initial states for the
solution of the time-dependent equations describing the electron density dynamics in the
system in response to the impulsive perturbation. For the“overcharged” clusters with
negative charge Q < Qcr the computational mesh has been restricted to r < Rext which
allows to stabilize the system and avoid the population decay.
In Fig. 6.5 we present the TDDFT results for the the time evolution of induced dipoles
and optical response of the charged Na [panels a) and c)] and Ag [panels b) and d)]
clusters. As follows from our results, the charging of the cluster affects both the energy
and width of the plasmon resonance, albeit in a very moderate way as has been also
reported in Ref. [178]. Stronger effects are produced (i) by the negative charges, and
(ii) for the Na cluster with lower work function and thus larger electron density spill out
into the vacuum. In general, the removal of the electrons from the cluster (positively
charged clusters) leads to the blue shift of the plasmon resonance frequency, and the
plasmon resonance becomes narrower and better defined. Adding electrons to the cluster
(negatively charged clusters) leads to the red shift of the plasmon resonance. These
results are in full accord with earlier experimental and theoretical data reported in
108
6.4
Ag Cluster
Induced dipole
Induced dipole
Na Cluster
b)
Na
Cluster
Ag
Cluster
σ(nm 2 )
a)
σ(nm 2 )
Absorption cross-section
d)
c)
Figure 6.5: TDDFT calculations of the optical response of the charged clusters.
Panels a) (sodium) and b) (silver) show the time evolution of the dipole induced in the
charged cluster by the impulsive perturbation. Panels c) and d) show the corresponding
absorption spectra. Different colors are used for the results obtained with different
cluster charges as indicated in the insets.
cluster physics [195, 224–229], and are opposed to the assumptions frequently made in the
explanation of the photochemical data, where negative/positive charge on the cluster is
supposed to blue/red shift the plasmon resonance through the overall increase/decrease
of the electron density n.
As has been shown in the theory of dynamic screening [56, 230–232], the shift of the
√
dipolar plasmon frequency from the classical Mie value given by ωp / 3 is proportional
to the position of the dynamically induced screening charges with respect to the cluster
boundary. For positively charged cluster the electrons are tighter bound in the strong
attractive potential well. The spill out of the electron density outside cluster boundaries
decreases leading to the blue shift of the dipolar plasmon frequency as compared to that
for the neutral cluster. At variance, for the negatively charged clusters, the electron density protrudes further away from the cluster boundaries because of the reduced binding
109
Chapter 6
so that the dipolar plasmon redshifts.
The broadening of the absorption resonance for the negatively charged clusters as seen
in the optical absorption cross section in Fig. 6.5c) and Fig. 6.5d) reflects faster decay
of the underlying plasmon mode, as clearly seen in the time domain dynamics of the
induced dipoles in Fig. 6.5a) and Fig. 6.5b). The increase of the decay rate is associated
with increased coupling between plasmon and continuum of the one-particle electron-hole
excitations, where along with the hot electron production inside cluster, the hot electron
produced by the plasmon decay can be emitted from the cluster [233]. Considering the
work function of silver (Φ = 4.65 eV) and sodium (Φ = 3. eV) clusters addressed in
this work, the dipolar plasmon decay with electron emission is impossible for silver,
and concerns only the electrons within ∼ 0.2 eV energy range from the Fermi level for
sodium cluster. However, the Fermi level EFQ of the negatively charged cluster (Q < 0)
approaches the vacuum level as expressed with Eq. 6.2. As a result, the probability
of the plasmon decay with electron emission grows. Alternatively, the positive charge
(Q > 0) the overall binding energy of the cluster electrons increases and these are tighter
confined within cluster boundaries.
6.5.
TDDFT study of the electron population decay in
negatively charged clusters
6.5.1.
Analytical expressions
As explained above, for the negatively charged clusters, when the negative charge exceeds
Qcr by q = Q − Qcr , the extra charge q decays because of the electron escape into the
continuum. Prior to the discussion of the results of the TDDFT studies of this effect
let us outline some simple theoretical considerations which ease understanding of the
numerical data. The WentzeI-Kramers-Brillouin (WKB) approximation allows reliable
estimation of the decay rate of the population of the cluster electronic states close to the
Fermi level EF via resonant (energy conserving) tunneling through the potential barrier
located at Rc ≤ r ≤ Rext shown in Fig. 6.4. The probability of the barrier penetration
and so the decay rate Γ is proportional to Γ ∼ exp(−2γ), where the γ parameter is given
by:
Z
Rext
γ=
Rc
s |Q|
2
− EF dr.
r
110
(6.9)
6.5
TDDFT study of the electron population decay in negatively charged clusters
Here we made an approximation assuming that the −Q/r dependence of the potential
acting on tunneling electron holds over the entire classically forbidden region Rc ≤ r ≤
Rext . This allows to use the derivations well developed in the theory of alpha-decay
[234]. Eq. 6.9 can be rearranged into the form
Rext
Z
p
γ = 2EF
s
Rc
Using
|Q|
EF
|Q| 1
− 1 dr.
EF r
(6.10)
= Rext [Eq. (6.5)], we finally obtain the main working equation defining the
penetration constant.
Z
p
γ = 2EF
Rext
r
Rc
Rext
− 1 dr.
r
(6.11)
Since r ≤ Rext , we can use the variable change r = Rext sin2 (ψ) and transform Eq. (6.11)
to:
γ = 2Rext
Z
p
2EF
π/2
h√
i cos
arcsin
Rc /Rext
2
(ψ) dψ
(6.12)
Finally, we obtain the sought penetration factor γ as
γ = Rext
h
p
p
i
p
p
2EF π/2 − arcsin
Rc /Rext − Rc /Rext 1 − Rc /Rext ,
(6.13)
where the external radius Rext is given by eq. (6.6).
Analytical derivation given by Eq. 6.13 allows discussion of the threshold behaviour of
the population decay for EF → 0+ , where the decay rate is small and it is difficult to
obtain from the full quantum TDDFT calculations. In this case the cluster charge is
given by Q = Qcr + q (q → 0− ), and Rc /Rext << 1 [see Eq. 6.5]. Using asymptotic
expansion we obtain
γ = Rext
p
h
i
p
p
2EF π/2 − Rc /Rext − (1 − Rc /2Rext ) Rc /Rext .
(6.14)
Keeping the terms of the same order with respect to Rc /Rext leads to
γ = Rext
h
i
p
p
2EF π/2 − 2 Rc /Rext .
(6.15)
Since EF = |Q|/Rext Eq. (6.15) further simplifies to
γ=
h
i
p
p
2QRext π/2 − 2 Rc /Rext .
111
(6.16)
Chapter 6
For Rext → ∞ the leading term is
π p
QRext .
γ=√
2
(6.17)
Recaling that the external radius, is given by Eq. 6.6 and taking Q = Qcr + q, we obtain
for Rext
Rext =
Qcr
Rc .
q
and
π
γ = √ |Qcr |
2
Rc
|q|
(6.18)
1/2
.
(6.19)
Therefore, since the charge population decay rate is given by Γ = Γ0 exp [−2γ], we obtain
"
√
Γ = Γ0 exp − 2π |Qcr |
Rc
|q|
1/2 #
,
(6.20)
where |Qcr | = ΦRc , is the difference between the total cluster charge and the threshold
charge for the onset of decay Qcr , and Γ0 is a parameter that can be determined comparing predictions of Eq. 6.20 and TDDFT results.
6.5.2.
TDDFT results
In the TDDFT studies of the electron population decay for the negatively charged clusters, the ψj (~r) orbitals obtained in the DFT ground state calculations on restricted-size
mesh are used as an initial states for the time propagation. However, in difference to the
calculation of the absorption cross-section, the confinement constraint has been lifted
off at the time moment t = 0 by providing the mesh size extending to r Rext . The
negative charge of the cluster can then decay via electron escape into the vacuum. In
Fig. 6.6.a) and in Fig. 6.6.b) we show the the time evolution of the cluster charge Q(t)
or of the excess charge defined as q(t) = Q(t) − Qcr calculated for Na and Ag clusters.
For the highly charged clusters (Q ∼ −50), about 20 electrons are lost in less then 2000
a.u. of time (50 fs). The cluster charge drops, and the population decay slows down
approaching time constants characteristic for the low initial charges. Indeed, the smaller
is the cluster excess charge |q(t)|, the higher is the potential barrier separating potential
well inside cluster from vacuum region. This results in the lower population decay rate
Γ which is then a function of the time-dependent excess charge.
112
TDDFT study of the electron population decay in negatively charged clusters
Na Cluster
Charge (a. u.)
Charge (a. u.)
6.5
b)
Na Cluster
d)
Na Cluster
Decay rate Γ (a. u.)
c)
Ag Cluster
a)
Na Cluster
Ag Cluster
f)
e)
g)
Ag Cluster
Ag Cluster
h)
q-1/2 (a. u.)
q-1/2 (a. u.)
Figure 6.6: TDDFT analysis of the negative charge decay in small sodium (left
column) and silver (right column) negatively charged clusters. Panels a) and b) timeevolution of the cluster charge Q(t) (solid lines) and excess charge above the decay
threshold q(t) = Q(t) − Qcr (dashed lines). Absolute values of the charges are shown.
Different colors are used to display the results obtained with different initial charge
Q(t = 0) (excess charge q(t = 0)) as explained in the inserts. t = 0 is the instant of
time when the computational constraints are released and system is allowed to decay.
The threshold charge |Qcr | is shown in black dashed line at the bottom of the panels.
Panels c) and d): the time-dependence of the excess charge population decay rates as
extracted from the data presented in panels a) and b) of this figure. Different colors are
used to represent the results obtained with different initial cluster charge s explained
in the inserts. Panels e) and f): the same as c) and d) but the results are shown
as function of the instantaneous cluster charge |Q(t)| Panels g) and h): analysis of
the validity of the analytical approach:
the logarithm of the instantaneous decay rate
p
ln(Γ(t)) is shown as function of 1/|q(t)|. for different values of the initial cluster
charge.
113
Chapter 6
The analysis of the population decay rate Γ is presented in panels c) - h) of Fig. 6.6.
Assuming the exponential decay of the excess charge, we introduce the instantaneous
decay rate Γ(t):
dq(t)
= −Γ(t)q(t).
dt
(6.21)
1 dq(t)
.
q(t) dt
(6.22)
From Eq. 6.21 it follows that
Γ(t) = −
We use this equation to obtain Γ(t) from the charge dynamics calculated with TDDFT
(see Fig 6.6.c) and Fig 6.6.d). The decay rate is in overall slower for low initial charges
and it also slows down in time as soon as the cluster becomes less charged. When the decay rate Γ is plotted as function of the instantaneous charge (see Fig 6.6.e) and Fig 6.6.f)
results obtained with different initial cluster charges Q(t = 0) fall into unique universal
curve. Thus, the potential barrier for electron tunneling and thus the decay rate is determined by the (instantaneous) cluster charge. The features at the short propagation
times arise because the exponential decay needs some time to set in as known from the
studies of the decaying quasi-stationary states [235].
In Fig 6.6.g) and Fig 6.6.h) the ln (Γ) is shown as function of the square root of the
p
excess charge |q(t)|. The same linear dependence is obtained irrespective from the
initial cluster charge. This is inline with prediction of the semiclassical treatment given
by Eq. 6.20. In e.g. case of silver clusters, the fit to the TDDFT results assuming
"
B
Γ = Γ0 exp − p
|q|
#
(6.23)
gives (in atomic units) B = 189 which matches extremely well the
"
178
Γ = Γ0 exp − p
|q|
#
(6.24)
dependence obtained from Eq. 6.20 with Rc = 38.16 a0 and Qcr = 6.52 a.u. Similar good
agreement is obtained for Na clusters, where B=207 as obtained from the linear fit to
√
the results in Fig 6.6.g has to be compared with 2Rc π |Qcr | = 176. With the validity
of the semiclassical approach established by comparison with TDDFT results, we can
use Eq. 6.20 to analyse the evolution of the population decay rates for the clusters of
3/2
varying size. Since the exponential function is proportional to Rc /|q|1/2 the population
decay rate can be expected to be similar for the charged clusters with |q| ∝ Rc3 . Since
114
6.6
Summary and Conclussions
the number of electrons in the cluster Ne is also proportional to the cluster volume,
we obtain that clusters characterized by q/Ne = const should have similar time scales
for the population relaxation. Using results shown in Fig 6.6a) and in Fig 6.6b) we
conclude that half of the excess charge representing 2 % of the total number of electrons
will be lost in 50 fs following the charging event. Thus, addition of small amount of
electron representing only some percent of the total amount of the valence electrons in
the nanoparticle renders the system highly unstable.
6.6.
Summary and Conclussions
Using the TDDFT calculations on the charged clusters we have demonstrated in this
chapter that the surface plasmon modes of small spherical clusters are only mildly sensitive to electron doping and do not present significant frequency shifts as has been also
found in Ref. [178]. This is because the electron density in the cluster volume can not
be changed and the excess charge is accumulated at the surface of the cluster.
In sheer contrast with an idea widely spread in the electrochemistry community, we
show that the negative charge leads to the red shift of the plasmon frequency and the
positive charge leads to the blue shift of the plasmon frequency. This result can be
understood within the theory of the dynamical screening as a consequence of the change
of the electron spill out from the cluster boundaries.
We have also shown that even small (relative to the total number of electrons in the system) negative charge rises Fermi level of the cluster above the vacuum level and renders
the system unstable. The extra charge of the cluster decays with characteristic time
scales of some tens of fs that can be fully understood using an analytical study based on
the WKB method, similar to the studies of the alpha-decay. This limits the possibility
to charge the system and the larger is the cluster, the smaller amount of the negative
charge Q it can admit relative to the total number of electrons in the cluster Ne . Our
results indicate that |Q|/Ne < 1%.
Our results on the charged clusters allow to critically reinterpret many of the experimental
data obtained with electrochemistry, where the shift of the plasmon resonances has been
115
Chapter 6
often attributed to the volume charging effects [206–209]. Based on our results, we tentatively propose an alternative explanation where the change of the dielectric environment
in immediate vicinity of the cluster should be responsible for the observed shift of the
plasmon fequency, as has been also conjectured in Refs. [236–238]
116
Chapter 7
General Summary
The main aim of this chapter is to summarize the most important contents and original
contributions made in this thesis.
7.1.
Analytical and numerical methods
In this thesis work, we have addressed several analytical and numerical methods to
describe the optical response of sub-nanometric systems. These methods are listed as
follows:
1. Classical electrodynamics and solution of Maxwell’s equations in metallic systems.
Finite Element Method (FEM):, implemented in the COMSOL Multyphysics package [34], which allows to find approximate solutions to systems of
partial differential equations (Maxwel’sl equations) describing the interaction
of light with metal systems. In our case, we use this powerful numerical tool
to study systems of tens of nanometers size.
117
Chapter 7
General Summary
Boundary Element Method (BEM):, implemented in a home-made code1 ,
basically permits to solve Maxwell’s equations formulated as integral equations. We use this method in plasmonic extended gaps, where a high symmetry is desirable.
2. Quantum methods to solve metallic systems in the nano and sub-nanometric
regime.
Time-Dependent Density Functional Theory (TDDFT): this formalism allows a quantum description of the density dynamics of hundreds to
thousands of electrons in metallic nano-systems represented by a free-electron
model (jellium). One can thus address nanoparticles, clusters or even hybrid
nanostructures subjected to an external time-dependent potential2 .
Quantum Corrected Model (QCM): within this model, it is possible to
describe the conductivity of a subnanometric cavity treating the junction as
an effective medium, mimicking quantum tunneling within the classical local
dielectric theory.
7.2.
Studied Systems
We used the methods mentioned above to investigate the optical properties, confinement,
tunneling process and in general the plasmonic behaviour of the following systems:
1. Single nanoparticles
Where we have studied:
Confinement potential effects on the optical response of small metal nanoparticles.
Finite size effects in subnanometric spherical particles.
2. Core-shell systems (cylindrical and spherical nanomatryoshkas) and plasmonic dimers
In these hybrid nanotructures we have studied:
1
developed by Javier Aizpurua from the Theory of Nanophotonics Group at Materials Physics Center,
Donostia-San Sebastian, España and Javier Garcı́a de Abajo from nanophotonics theory group at Institut
de Ciènces Fotòniques
2
The codes used to this purpose, where gently shared by Andrei Borisov and Codruta Marinica from
the Institut des Sciences Molèculaires d’Orsay, Universite Paris Sud, France.
118
7.3
Main results
The physics of the plasmon coupling and the near-field enhancements in the
plasmonic cavities formed in their junctions.
The regime where quantum effects like tunneling become important.
A mechanism of active control of the conductive coupling in plasmonic gaps
between metallic nanoparticles via the bias-induced change of the barrier for
electron tunneling across the gap.
3. Small charged cluster
We have performed full quantum calculations of the electron dynamics in small
spherical charged clusters in order to study:
The optical response and the frequency shifts of the plasmon resonances as a
function of the cluster charge.
Stability and charge excess decay in negatively charged clusters and the correspondence with analytical studies based on the WKB method.
7.3.
Main results
The following original results are obtained in this thesis.
1. We have demonstrated the applicability of the quantum corrected model to the
description of the optical properties and plasmonic modes in systems which show
an extended tunneling contact region between metallic surfaces.
2. We obtained that when the core-shell gap (in Na nanomatryoshkas) is reduced
below 5 Å, the optical response is determined by the quantum tunneling of conduction electrons across the potential barrier separating the core and the shell.
Our results agree with earlier calculations on canonical systems like plasmonic
dimers.
3. We have demonstrated that the quantum corrected model performs very well over
a wide range of core-shell gap sizes and geometries.
4. Thus, we could apply the quantum corrected model to describe the experimental
findings in gold nanomatryoshkas of tens of nm size, where full quantum calculations are currently not feasible. From the comparison between the experimental
and theoretical data we could show that, in case of subnanometric gaps between
119
Chapter 7
General Summary
the core and the shell functionalised with self-assembled molecular layer(s), the
tunneling through the latter determines the optical response of the nanostructure.
5. With TDDFT and QCM calculations we have demonstrated that the optical
response of a plasmonic gap can be electrically controlled by an external DC bias.
We could elucidate the mechanism underlying this effect which is the bias induced
change of the electron tunneling conductance across the gap.
6. Our proof-of -principle calculations show that metal nanoparticle plasmonics can
benefit from the reversibility, fast response time, and versatility of an active control
strategy based on applied bias. This provides a new platform for many practical
applications in optoelectronics and photonics.
7. Using TDDFT calculations we have demonstrated that the localised plasmon
modes of small spherical clusters are only mildly sensitive to electron doping and
do not present significant frequency shifts. Consistently with the theory of dynamical screening at surfaces, and in contrast to wide spread ideas, the negative
charge of the cluster leads to a small red shift of the plasmon frequency and the
positive charge leads to a small blue shift of the plasmon frequency.
8. We have demonstrated that even small, negative charge of the cluster, relative to
the total number of electrons in small charged systems, rises the Fermi level above
the vacuum level and renders the system unstable. The extra charge decays with
characteristic time scales of some tens of fs. This, in fact, limits the possibility
to charge the system. The larger the cluster is, the smaller amount of negative
charge Q it can admit relative to the total number of electrons in the cluster Ne .
Overall, |Q|/Ne < 1%.
9. Our results on the charged clusters allow to critically reinterpret many of the
experimental data obtained with electrochemistry, where the shift of the plasmon
resonances has been often attributed to charging effects. Based on our results, we
tentatively propose an alternative explanation based on the change of dielectric
environment in immediate vicinity of the clusters as the main reason for plasmon
shifts in electrochemical environments.
120
Appendix A
Quantum model for a free
electron gas in a nanoparticle
A.1.
Free electron gas in a cubic infinite box
In a simplified quantum model described by Genzel and Martin in [26], which is based
on a set of free (valence) electrons confined in a box or cubic potential well with infinite
sides, the electron wave function is well known and given by
Ψhkl =
8
L3
sin
hπx
L
sin
kπy
L
sin
lπz
L
,
(A.1)
where L is the lenght of the potential well and the h, k, l are positive integers. For each
wave function, the corresponding energy levels are given by
Eh,k,l = E0 (h2 + k 2 + l2 )
with E0 =
π 2 ~2
.
2mL2
(A.2)
In this model, the dielectric function of a single metal nanoparticle is
given by
m
ωp2 X
sif (Fi − Ff )
ε(ω) = ε∞ +
,
2
N
ωif − ω 2 − iωγif
(A.3)
i,f
where ε∞ corresponds to the interband contribution, N to the number of particles
confined in the particle and the plasmon frequency ωp is given in terms of the electronic
121
Appendix A.Potentials
density n, the elemental electric charge e and the effective mass m∗ ,
ωp2 =
4πne2
.
m∗
(A.4)
The terms sif , ωif and γif are respectively, the oscillator strenght, the eigen frequency
and the damping for the dipole transition from a initial state i to another final state f .
Fi and Ff are the values of the Fermi-Dirac distribution function for the initial and final
states. The oscillator strenght has the standard dipolar form
sif =
2mωif
|hf |z|ii|2 .
~
(A.5)
The orthogonallity of the eigen-functions, gives the relation
ωif = ∆l(2l + ∆l)
E0
~
(A.6)
and considering only transitions in a z direction, sif can be written by
s2if =
64 l2 (l + ∆l)2
δ(hi , hf )δ(ki , kf ).
π 2 ∆l3 (l + ∆l)3
(A.7)
The number of dipole transitions in the sum of the dielectric function depends on the
initial and final values of the quantum number l corresponding solely to initial states
within the Fermi surface through final states just outside the Fermi surface. Considering
the number of valence electrons in each nanoparticle N and taking only transitions with
∆l = 1, the final expression obtained in [26] reads
(ω) = ∞ + ωp2
1,3,...
lF
XX
∆l
l
2
ωl,∆l
Sl,∆l
− ω 2 − iωγl,∆l
(A.8)
where
Sl,∆l =
32 l2 (l + ∆l)2
πlF ∆l2 (l + ∆l)2
(A.9)
In other derivation made by Scholl et. al. in [55], the conduction electrons are considered
as a free electron gas constrined by an infinite potential barriers at the physical edges of
the particles. Details are showed in Chapter 2 They take the transition frequencies ωif
as the corresponding to the transitions of conduction electrons from occupied states i
to unocupied states f inmediately outside of it. The dielectric function ε(ω) is given in
terms of the constant plasmon bulk frequency ωp , the damping (or scattering) frequency
122
A.1
Figure A.1: a) Real and b) imaginary parts of the dielectric function for a silver
nanoparticle of lenght L = 1.6, 3.6, 4.8, 6.4, 8 and 16 nm. Using an infinite cubic confinement potential.
dependent of the particle radius γ(R) = γbulk + AvF /R and the oscillator strenghts Sif
in the form
ε(ω) = εIB + ωp2
XX
i
f
2
ωif
Sif
.
− ω 2 − iγω
(A.10)
In this expression, Sif and ωif are given respectively by
Sif =
2M ωif
|hf |z|ii|2 .
~N
(A.11)
Where N is the number of conduction electrons in the nanoparticle and
ωif =
Ef − Ei
.
~
(A.12)
If the conduction electrons are treated as particles in a infinite spherical well, the energy
eigen-values will depend on the quantum numbers n (principal) and l (azimuthal) in the
form:
En,l =
~2 π 2
(2n + l + 2)2 ,
8M R2
(A.13)
with M the mass of the electron. Using Eq. (A.8), the real and imaginary parts of
the dielectric function in terms of the the incident photon energy for nanoparticles with
L = 1.6, 3.6, 4.8, 6.4, 8 and 16 nm (we can relate the lenght of the cube L with the radius
1/3 R), is showed in Fig. A.1 comparing with
of a nanosphere by the expresion L = ( 4π
3 )
the classic Drude approach given by eq. (1.2) in the case of L = 16nm. We observe
123
Figure A.2: Absorption spectra for a silver nanoparticle using the optical constants
from the quantum model: Infinite cubic potential (a and b) and infinite spherical
potential (c and d) for different particle sizes (L=1.6, 3.2, 4.8, 6.4, 8 and 16 nm and
R=1, 2, 3, 4, 5 nm and 10 nm). The graphs a) and c) correspond to a refractive index
of surrounding medium n = 1.0 and the graphs b) and d) to n = 1.5.
that the dielectric functions predicted by the quantum model disagree notably with the
classical results in the range studied. The main difference is that both Re[] and Im[]
exhibit a nonmonotic behavior according to the quantum model, versus the monotonic
response using classical Drude theory. However, as the lenght reaches the limit of 16
nm, the agreement with the classical description is good.
To solve the problem of light-nanoparticle interaction, we use the analytical expressions
of η and κ previously calculated for each particle size and confinement potential to solve
the vector Maxwell equations numerically, by means of a standar finite element method
(FEM). For an isolated silver particle with different sizes (R = 1, 2, 3, 4, 5 and 10 nm),
we compare the solutions for the different sets of optical constants, which are taken as
input parameters.
124
A.1
Figure A.2, shows the resonance value from each absorption spectra for both confinement potential (cubic and spherical), depending on the surrounding environment: air,
with refractive index n = 1.0 and SiO2 (glass) with n = 1.5, in which we observe that
the LSPR in both spectra change significantly with this parameter, showing that in the
glass case, the plasmon resonance values are more defined and follow a well marked
tendency relative to real experiments (see [55] and [56]).
125
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[235] Nakazato, H.; Namiki, M.; Pascazio, S., “Temporal Behaviour of Quantum Mechanical Systems”, Int. J. Mod. Phys. B 10, 247 (1996)
[236] Daniels, J. K.; Chumanov, G., “Spectroelectrochemical studies of plasmon coupled
silver nanoparticles”, Journal of Electroanalytical Chemistry 575, 203–209 (2005)
[237] König, T. A. F.; Ledin, P. A.; Kerszulis, J.; Mahmoud, M. A.; El-Sayed, M.
A.; Reynolds, J. R.; Tsukruk, V. V., “Electrically Tunable Plasmonic Behavior
of Nanocube–Polymer Nanomaterials Induced by a Redox-Active Electrochromic
Polymer”, ACS Nano 8, 6182-6192 (2014)
[238] Mogensen, K. B.; Kneipp, K., “Size-Dependent Shifts of Plasmon Resonance in
Silver Nanoparticle Films Using Controlled Dissolution: Monitoring the Onset of
Surface Screening Effects”. J. Phys. Chem. C 118, 28075–28083 (2014)
147
Abbreviations
AC
Alternating Current
BEM
Boundary Element Method
CNM
Cylindrical Nano Matryoshka
CT
Charge Transfer
DC
Direct Current
DM
Drude Model
EFF-EF
Enhancement Field Factor
FEM
Finite Element Method
HDM
Hydro Dynamic Model
JM
Jellium Model
LSP
Localized Surface Plasmon
LSPP
Localized Surface Plasmon Polariton
LSPR
Localized Surface Plasmon Resonance
ME
Maxwell Equations
NM
Nano Matryoshka
NP
Nano Particle
NPD
Nano Particle Dimer
PH
Plasmon Hybridization
QCM
Quantum Corrected Model
SERS
Surface Enhanced Raman Spectroscopy
TDDFT
Time Dependent Density Functional Theory
WKB
Wentzel Kramers Brillouin Approximation
149
The articles published (or in preparation) during the period of this thesis are listed
below:
Regular papers
1. “Quantum effects in the optical response of extended plasmonic gaps: validation of
the quantum corrected model in core-shell nanomatryushkas” Zapata, M., Beltran,
A. S. C., Borisov, A. G., Aizpurua, J. Opt. Express 2015,23, 8134-8149.
2. “Nanooptics of Plasmonic Nanomatryoshkas: Shrinking the Size of a Core-Shell
Junction to Subnanometer.” Lin. L, Zapata. M, Xiong. M, Liu. Z, Wang. S, Xu.
H, Borisov. A. G., Gu. H, Nordlander. P, Aizpurua. J, and Jian Ye. Nano Lett.,
2015, 15 (10), pp 6419–6428
1
3. “Active Quantum Plasmonics.” D. C. Marinica, M. Zapata, P. Nordlander, A. K.
Kazansky, P. M. Echenique, J. Aizpurua, A. G. Borisov. Sci. Adv.1, e1501095
(2015).
4. “Quantum Plasmonics and charged cluster dynamics.” Zapata. M, Borisov. A.
G, Kazanski. A. K. and Aizpurua. J. Submitted to Langmuir.
5. “Size change effect on the optical behavior of ultra small metal particles.” Mario
Zapata Herrera and Angela S. Camacho B. In preparation.
1
Contribution equally to this work as first author.
151
6. “Quantum confinement effects on the dielectric function and near field enhancement in metallic nanoparticles”. M. Zapata Herrera, J. Florez, A. S. Camacho and
H. Y. Ramirez. In preparation.
Divulgative papers:
1. “Plasmónica en el régimen subnanométrico”. Angela Camacho y Mario Zapata
Herrera. Revista Digital Universitaria. 1 de septiembre de 2015, Vol. 16, Núm. 9.
http://www.revista.unam.mx/vol.16/num9/art72/
Conferences
1. “Study of the geometrical dependence of plasmonics resonances Using FiniteDifference Time-Domain Method (FDTD)”. M. Zapata, J. C. Arias, A. Camacho
and H. Garcia. XII Encuentro Nacional de Óptica y III Conferencia Andina y
del Caribe en Óptica y sus Aplicaciones. Universidad del Atlántico, Barranquilla.
September 5-9/2011.
2. “Artificial Plasmonic Molecules”. M. Zapata Herrera and A. Camacho. Nanosmat
USA 2012. Tampa, Florida USA. March 2012.
3. “Quantum plasmon resonances and coupling of small nanoparticles”. M. ZapataHerrera and A. Camacho. APS March Meeting 2013. Baltimore, Maryland, USA.
March 2013.
4. “Localized Surface Plamons in very small nanoparticles”. 14th Conference on
Physics of Light-Matter Coupling in Nanostructures. Hersonissos, Crete, Greece.
May 2013.
5. “Effect of the confinement potential on the Localized Surface Plasmon Resonance
of metallic small nanoparticles”. M. Zapata Herrera, J. Florez and A. Camacho.
Semana de la Nanociencia y la Nanotecnologı́a en Colombia Nanoantioquia 2013.
Medellin, Colombia. Julio de 2013.
152
ABBREVIATIONS
6. “Finite potential and nonlinear contribution effects in the Localized Surface Plasmon Resonances (LSPR) of small metallic nanoparticles”. M. Zapata Herrera and
A. Camacho. XXV Congreso nacional de Fı́sica. Armenia, Colombia. Agosto de
2013.
Scientific visits
Centro de Fı́sica de Materiales and Donostia International Physics Center, Universidad del Paı́s Vasco, Spain. May-August 2014.
Universitè Paris Sud, France. September-December 2014.
Centro de Fı́sica de Materiales and Donostia International Physics Center, Universidad del Paı́s Vasco, Spain. May-August 2015.
Centro de Fı́sica de Materiales and Donostia International Physics Center, Universidad del Paı́s Vasco, Spain. November-December 2015.
153

Thesis Manuscript - Materials Physics Center

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